An Exp erimental Approach to Nonlinear Dynamics
and Chaos
by
Nicholas B Tu llaro Tyler Abb ott and Jeremiah P Reilly
INTERNET ADDRESSES nbt reed edu
c
COPYRIGHT TUFILLARO ABBOTT AND REILLY
Published by Addison Wesley
v b
Novemb er
i
ii
iii
iv
Contents
Preface xi
Intro duction
What is nonlinear dynamics
What is in this b o ok
Some Terminology Maps Flows and Fractals
References and Notes
Bouncing Ball
Intro duction
Mo del
Stationary Table
Impact Relation for the Oscillating Table
The Equations of Motion Phase and Velo city
Maps
Parameters
High Bounce Approximation
Qualitative Description of Motions
Trapping Region
Equilibrium Solutions
Sticking Solutions
Perio d One Orbits and Perio d Doubling
Chaotic Motions
Bifurcation Diagrams
References and Notes
Problems
Quadratic Map
Intro duction
Iteration and Di erentiation
Graphical Metho d
Fixed Points v
vi CONTENTS
Perio dic Orbits
Graphical Metho d
Perio d One Orbits
Perio d Two Orbit
Stability Diagram
Lo cal Bifurcation Theory
Saddle no de
Perio d Doubling
Transcritical
Perio d Doubling Ad In nitum
Sarkovskii s Theorem
Sensitive Dep endence
Fully Develop ed Chaos
Hyp erb olic InvariantSets
Symb olic Dynamics
Top ological Conjugacy
Symb olic Co ordinates
What s in a name Lo cation
Alternating Binary Tree
Top ological Entropy
Usage of Mathematica
References and Notes
Problems
String
Intro duction
Exp erimental Apparatus
Single Mo de Mo del
Planar Vibrations Du ng Equation
Equilibrium States
Unforced Phase Plane
Extended Phase Space
Global Cross Section
Resonance and Hysteresis
Linear Resonance
Nonlinear Resonance
Resp onse Curve
Hysteresis
Basins of Attraction
Homo clinic Tangles
Nonplanar Motions
Free Whirling
Resp onse Curve
CONTENTS vii
Torus Attractor
Circle Map
Torus Doubling
Exp erimental Techniques
Exp erimental Cross Section
Emb edding
Power Sp ectrum
Attractor Identi cation
References and Notes
Problems
Dynamical Systems Theory
Intro duction
Flows and Maps
Flows
Poincar e Map
Susp ension of a Map
Creed and Quest
Asymptotic Behavior and Recurrence
InvariantSets
Limit Sets and Nonwandering
Chain Recurrence
Expansions and Contractions
DerivativeofaMap
Jacobian of a Map
Divergence of a Vector Field
Dissipative and Conservative
Equation of First Variation
Fixed Points
Stability
Linearization
Hyp erb olic Fixed Points Saddles Sources and
Sinks
Invariant Manifolds
Center Manifold Theorem
Homo clinic and Hetero clinic Points
Example Laser Equations
Steady States
Eigenvalues of a Matrix
Eigenvectors
Stable Focus
Smale Horsesho e
From Tangles to Horsesho es
viii CONTENTS
Horsesho e Map
Symb olic Dynamics
From Horsesho es to Tangles
Hyp erb olicity
Lyapunov Characteristic Exp onent
References and Notes
Problems
Knots and Templates
Intro duction
Perio dic Orbit Extraction
Algorithm
Lo cal Torsion
Example Du ng Equation
Knot Theory
Crossing Convention
Reidemeister Moves
Invariants and Linking Numb ers
Braid Group
Framed Braids
Relative Rotation Rates
Templates
Motivation and Geometric Description
Algebraic Description
Lo cation of Knots
Calculation of Relative Rotation Rates
Intertwining Matrices
Horsesho e
Lorenz
Du ng Template
References and Notes
Problems
App endix A Bouncing Ball Co de
App endix B Exact Solutions for a Cubic Oscillator
App endix C Ode Overview
App endix D Discrete Fourier Transform
App endix E H enon s Trick
App endix F Perio dic Orbit Extraction Co de
CONTENTS ix
App endix G Relative Rotation Rate Package
App endix H Historical Comments
App endix I Pro jects
x CONTENTS
Preface
An Experimental Approach to Nonlinear Dynamics and Chaos is a textb o ok and a
reference work designed for advanced undergraduate and b eginning graduate stu
dents This b o ok provides an elementary intro duction to the basic theoretical and
exp erimental to ols necessary to b egin researchinto the nonlinear b ehavior of me
chanical electrical optical and other systems A fo cus of the text is the description
of several desktop exp eriments such as the nonlinear vibrations of a current carrying
wire placed b etween the p oles of an electromagnet and the chaotic patterns of a ball
b ouncing on a vibrating table Each of these exp eriments is ideally suited for the
small scale environment of an undergraduate science lab oratory
In addition the b o ok includes software that simulates several systems describ ed
in this text The software provides the student with the opp ortunity to immedi
ately explore nonlinear phenomena outside of the lab oratory The feedbackofthe
interactive computer simulations enhances the learning pro cess by promoting the
formation and testing of exp erimental hyp otheses Taken together the text and
asso ciated software provide a hands on intro duction to recent theoretical and ex
p erimental discoveries in nonlinear dynamics
Studies of nonlinear systems are truly interdisciplinary ranging from exp erimen
tal analyses of the rhythms of the human heart and brain to attempts at weather
prediction Similarly the to ols needed to analyze nonlinear systems are also inter
disciplinary and include techniques and metho dologies from all the sciences The
to ols presented in the text include those of
theoretical and applied mathematics dynamical systems theory and p erturba
tion theory
theoretical physics development of mo dels for physical phenomena application
of physical laws to explain the dynamics and the top ological characterization
of chaotic motions
exp erimental physics circuit diagrams and desktop exp eriments
engineering instabilities in mechanical electrical and optical systems and
computer science numerical algorithms in C and symb olic computations with
Mathematica xi
xii
A ma jor goal of this pro ject is to showhowtointegrate to ols from these di erent
disciplines when studying nonlinear systems
Many sections of this b o ok develop one sp eci c to ol needed in the analysis
of a nonlinear system Some of these to ols are mathematical such as the appli
cation of symb olic dynamics to nonlinear equations some are exp erimental such
as the necessary circuit elements required to construct an exp erimental surface of
section and some are computational such as the algorithms needed for calculat
ing fractal dimensions from an exp erimental time series We encourage students to
try out these to ols on a system or exp erimentoftheirown design To help with
this App endix I provides an overview of p ossible pro jects suitable for researchby
an advanced undergraduate Some of these pro jects are in acoustics oscillations
in gas columns hydro dynamics convective lo op Lorenz equations Hele Shaw
cell surface waves mechanics oscillations of b eams stability of bicycles forced
p endulum compass needle in oscillating B eld impact oscillators chaotic art mo
biles ball in a swinging track optics semiconductor laser instabilities laser rate
equations and other systems showing complex b ehavior in b oth space and time
video feedback ferrohydro dynamics capillary ripples
This b o ok can b e used as a primary or reference text for b oth exp erimental
and theoretical courses For instance it can b e used in a junior level mathematics
course that covers dynamical systems or as a reference or lab manual for junior
and senior level physics labs In addition it can serve as a reference manual for
demonstrations and p erhaps more imp ortantly as a source b o ok for undergraduate
research pro jects Finally it could also b e the basis for a new interdisciplinary
course in nonlinear dynamics This new course would contain an equal mixture of
mathematics physics computing and lab oratory work The primary goal of this
new course is to give students the desire skills and con dence needed to b egin their
own researchinto nonlinear systems
Regardless of her eld of study a student pursuing the material in this b o ok
should have a rm grounding in Newtonian physics and a course in di erential
equations that intro duces the qualitative theory of ordinary di erential equations
For the latter chapters a go o d dose of mathematical maturity is also helpful
To assist with this new course we are currently designing labs and software
including complementary descriptions of the theory for the b ouncing ball system
the double scroll LRC circuit and a nonlinear string vibrations apparatus The
b ouncing ball package has b een completed and consists of a mechanical apparatus
a loudsp eaker driven by a function generator and a ball b earing the Bouncing Ball
simulation system for the Macintosh computer and a lab manual This package
has b een used in the Bryn Mawr College Physics Lab oratory since
This text is the rst step in our attempt to integrate nonlinear theory with easily
accessible exp eriments and software It makes use of numerical algorithms symb olic
packages and simple exp eriments in showing how to attack and unravel nonlinear
problems Because nonlinear e ects are commonly observed in everyday phenomena
avalanches in sandpiles a dripping faucet frost on a window pane they easily
capture the imagination and more imp ortantly fall within the research capabilities
xiii
of a young scientist Many exp eriments in nonlinear dynamics are individual or
small group pro jects in which it is common for a studenttofollow an exp eriment
from conception to completion in an academic year
In our opinion nonlinear dynamics research illustrates the nest asp ects of small
science It can b e the e ort of a few individuals requiring mo dest funding and often
deals with homemade exp eriments which are intriguing and accessible to students
at all levels We hop e that this b o ok helps its readers in making the transition from
studying science to doing science
We thank Neal Abraham Al Albano and Paul Melvin for providing detailed
comments on an early version of this manuscript We also thank the Department
of Physics at Bryn Mawr College for encouraging and supp orting our e orts in this
direction over several years Wewould also like to thank the text and software
reviewers who gave us detailed comments and suggestions Their corrections and
questions guided our revisions and the text and software are b etter for their scrutiny
One of the b est parts ab out writing this b o ok is getting the chance to thank all
our co workers in nonlinear dynamics We thank Neal Abraham Kit Adams Al Al
bano Greg Alman Ditza Auerbach Remo Badii Richard Bagley Paul Blanchard
Reggie Brown Paul Bryant Gregory Buck Jose na Casasayas Lee Casp erson R
Crandall Predrag Cvitanovi c Josh Degani Bob Devaney Andy Dougherty Bonnie
Duncan Brian Fenny Neil Gershenfeld Bob Gilmore Bob Gioggia Jerry Gollub
David Gri ths G Gunaratne Dick Hall Kath Hartnett Doug Hayden Gina Luca
and Lois Ho er Lippi Phil Holmes Reto Holzner Xin Jun Hou Tony Hughes
Bob Jantzen Raymond Kapral Kelly and Jimmy Kenison Falkner Tim Kerwin
Greg King Eric Kostelich Pat Langhorne D Lathrop Wentian Li Barbara Litt
Mark Levi Pat Lo cke Amy Lorentz Takashi Matsumoto Bruce McNamara Tina
Mello Paul Melvin Gabriel Mindlin Tim Molteno Ana Nunes Oliver O Reilly
Norman Packard R Ramshankar Peter Rosenthal Graham Ross Miguel Rubio
Melora and Roger Samelson Wes Sandle Peter Saunders Frank Selker John Selker
Bill Sharpf Tom Shieb er Francesco Simonelli Lenny Smith Hern an Solari Tom
Solomon Vernon Squire John Stehle Rich Sup er ne M Tarro ja Mark Taylor J
R Tredicce Hans Troger Jim Valerio Don Warrington Kurt Wiesenfeld Stephen
Wolfram and Kornelija Zgonc
Wewould like to express a sp ecial word of thanks to Bob Gilmore Gabriel
Mindlin Hern an Solari and the Drexel nonlinear dynamics group for freely sharing
and explaining their ideas ab out the top ological characterization of strange sets We
also thank Tina Mello for exp erimental exp ertise with the b ouncing ball system
Tim Molteno for the design and construction of the string apparatus and Amy
Lorentz for programing exp ertise with Mathematica
Nickwould liketoacknowledge agencies supp orting his research in nonlinear
dynamics whichhave included Sigma Xi the FulbrightFoundation Bryn Mawr
College Otago University Research Council the Beverly Fund and the National
Science Foundation
Nickwould like to o er a sp ecial thanks to Mom and Dad for lots of home
co oked meals during the writing of this b o ok and to Mary Ellen and Tracy for
xiv
encouraging him with his chaotic endeavors from an early age
Tyler would like to thank his own family and the Duncan and Dill families for
their supp ort during the writing of the b o ok
JeremyandTyler would like to thank the exciting teachers in our lives Jeremy
and Tyler dedicate this b o ok to all inspiring teachers
Intro duction
What is nonlinear dynamics
A dynamical system consists of two ingredients a rule or dynamic
which sp eci es how a system evolves and an initial condition or state
from which the system starts The most successful class of rules for de
scribing natural phenomena are di erential equations All the ma jor
theories of physics are stated in terms of di erential equations This
observation led the mathematician V I Arnold to comment conse
quently di erential equations lie at the basis of scienti c mathematical
philosophy our scienti c world view This scienti c philosophy b egan
with the discovery of the calculus by Newton and Leibniz and continues
to the presentday
Dynamical systems theory and nonlinear dynamics grew out of the
qualitative study of di erential equations which in turn b egan as an
attempt to understand and predict the motions that surround us the
orbits of the planets the vibrations of a string the ripples on the
surface of a p ond the forever evolving patterns of the weather The rst
twohundred years of this scienti c philosophy from Newton and Euler
through to Hamilton and Maxwell pro duced many stunning successes
in formulating the rules of the world but only limited results in
nding their solutions Some of the motions around us such as the
swinging of a clo ck p endulum are regular and easily explained while
others such as the shifting patterns of a waterfall are irregular and
initially app ear to defy any rule
The mathematician Henri Poincar e was the rst to appreci
ate the true source of the problem the di cultylay not in the rules
but rather in sp ecifying the initial conditions At the b eginning of this
Intro duction
century in his essay Science and Method Poincar e wrote
Avery small cause which escap es our notice determines
a considerable e ect that we cannot fail to see and then
wesay that that e ect is due to chance If we knew ex
actly the laws of nature and the situation of the universe at
the initial moment we could predict exactly the situation
of that same universe at a succeeding moment But even
if it were the case that the natural laws had no longer any
secret for us we could still only know the initial situation
approximately If that enabled us to predict the succeed
ing situation with the same approximation that is all we
require and weshouldsay that the phenomenon had b een
predicted that it is governed bylaws But it is not always
so it may happ en that small di erences in the initial con
ditions pro duce very great ones in the nal phenomena A
small error in the former will pro duce an enormous error in
the latter Prediction b ecomes imp ossible and wehavethe
fortuitous phenomenon
Poincar e s discovery of sensitive dependence on initial conditions in
what are now termed chaotic dynamical systems has only b een fully
appreciated by the larger scienti c community during the past three
decades Mathematicians physicists chemists biologists engineers
meteorologists indeed individuals from all elds have with the help
of computer simulations and new exp eriments discovered for them
selves the cornucopia of chaotic phenomena existing in the simplest
nonlinear systems
Before we pro ceed we should distinguish nonlinear dynamics from
dynamical systems theory The latter is a well de ned branchof
mathematics while nonlinear dynamics is an interdisciplinary eld that
draws on all the sciences esp ecially mathematics and the physical sci
ences
For an outline of the mathematical theory of dynamical systems see D V
Anosov I U Bronshtein S Kh Aranson and V Z Grines Smo oth dynamical
systems in Encyclopaedia of Mathematical Sciences Vol edited by D V Anosov
and V I Arnold Springer Verlag New York
What is nonlinear dynamics
Scientists in all elds are united by their need to solve nonlinear
equations and each di erent discipline has made valuable contribu
tions to the analysis of nonlinear systems A meteorologist discov
ered the rst strange attractor in an attempt to understand the un
predictability of the weather A biologist promoted the study of the
quadratic map in an attempt to understand p opulation dynamics And
engineers computer scientists and applied mathematicians gaveusa
wealth of problems along with the computers and programs needed
to bring nonlinear systems alive on our computer screens Nonlinear
dynamics is interdisciplinary and nonlinear dynamicists rely on their
colleagues throughout all the sciences
To de ne a nonlinear dynamical system we rstlookatanexample
of a linear dynamical system A linear dynamical system is one in
which the dynamic rule is linearly prop ortional to the system variables
Linear systems can b e analyzed by breaking the problem into pieces
and then adding these pieces together to build a complete solution
For example consider the second order linear di erential equation
d x
x
dt
The dynamical system de ned by this di erential equation is linear b e
cause all the terms are linear functions of x The second derivativeof x
the acceleration is prop ortional to x Tosolve this linear di erential
equation wemust nd some function x t with the following prop erty
the second derivativeof x with resp ect to the indep endentvariable t
is equal to x Two p ossible solutions immediately come to mind
x t sin t and x t cos t
since
d
x t sin t x t
dt
and
d
x t cos t x t
dt
E N Lorenz Deterministic nonp erio dic ow J Atmos Sci
R M May Simple mathematical mo dels with very complicated dynamics Na
ture
Intro duction
that is b oth x and x satisfy the linear di erential equation Because
the di erential equation is linear the sum of these two solutions de ned
by
x t x t x t
is also a solution This can b e veri ed by calculating
d d d
x t x t x t
dt dt dt
x t x t
x t
Anynumb er of solutions can b e added together in this way to form a
new solution this prop erty of linear di erential equations is called the
principle of superposition It is the cornerstone from which all linear
theory is built
Now let s see what happ ens when we apply the same metho d to
a nonlinear system For example consider the second order nonlinear
di erential equation
d x
x
dt
Let s assume wecan ndtwo di erent solutions to this nonlinear dif
ferential equation whichwe will again call x t and x t A quick
calculation
d x d x d x
dt dt dt
x x
x x x x
x x
x
shows that the solutions of a nonlinear equation cannot usually b e
added together to build a larger solution b ecause of the cross terms
x x The principle of sup erp osition fails to hold for nonlinear sys
tems
The full de nition of a linear system also requires that the sum of scalar pro ducts
x t a x t a x t where a and a are constants is also a solution
What is nonlinear dynamics
Traditionally a di erential equation is solved by nding a func
tion that satis es the di erential equation A tra jectory is then de
termined by starting the solution with a particular initial condition
For example if wewant to predict the p osition of a comet ten years
from nowwe need to measure its current p osition and velo city write
down the di erential equation for its motion and then integrate the
di erential equation starting from the measured initial condition The
traditional view of a solution thus centers on nding an individual orbit
or tra jectory That is given the initial condition and the rule weare
asked to predict the future p osition of the comet Before Poincar e s
work it was thought that a nonlinear system would always haveaso
lution we just needed to b e clever enough to nd it
Poincar e s discovery of chaotic b ehavior in the three b o dy problem
showed that such a view is wrong No matter howclever wearewewon t
b e able to write down the equations that solvemany nonlinear systems
This is not wholly unexp ected After all in a b ounded closed form
solution we might exp ect that any small change in initial conditions
should pro duce a prop ortional change in the predicted tra jectories But
achaotic system can pro duce large di erences in the long term tra jec
tories even when two initial conditions are close Poincar e realized the
full implications of this simple discovery and he immediately rede ned
the notion of a solution to a di erential equation
Poincar ewas less interested in an individual orbit than in all p os
sible orbits He shifted the emphasis from a lo cal solution knowing
the exact motion of an individual tra jectory to a global solution
knowing the qualitativebehavior of all p ossible tra jectories for a given
class of systems In our comet example a qualitative solution for the
di erential equation governing the comet s tra jectory might app ear eas
ier to achieve since it would not require us to integrate the equations of
motion to nd the exact future p osition of the comet The qualitative
solution is often di cult to completely sp ecify though b ecause it re
quires a global view of the dynamics that is the p ossible examination
of a large numb er of related systems
Finding individual solutions is the traditional approach to solv
ing a di erential equation In contrast recurrence is a key theme in
Poincar e s quest for the qualitative solution of a di erential equation
To understand the recurrence prop erties of a dynamical system we
Intro duction
need to know what regions of space are visited and how often the orbit
returns to those regions We can seek to statistically characterize how
often a region of space is visited this leads to the so called ergo dic
theory of dynamical systems Additionally we can try to understand
the geometric transformations undergone by a group of tra jectories
this leads to the so called top ological theory of dynamical systems em
phasized in this b o ok
There are many di erentlevels of recurrence For instance the
comet could crash into a planet After that nothing much happ ens
unless you re a dinosaur Another p ossibility is that the comet could
go into an orbit ab out a star and from then on follow a p erio dic motion
In this case the comet will always return to the same p oints along the
orbit The recurrence is strictly p erio dic and easily predicted But there
are other p ossibilities In particular the comet could followa chaotic
path exhibiting a complex recurrence pattern visiting and revisiting
di erent regions of space in an erratic manner
To summarize Poincar eadvo cated the qualitative study of di er
ential equations Wemaylosesight of some sp eci c details ab out any
individual tra jectory butwewanttosketch out the patterns formed
by a large collection of di erent tra jectories from related systems This
global view is motivated by the fact that it is nonsensical to study
the orbit of a single tra jectory in a chaotic dynamical system To un
derstand the motions that surround us which are largely governed by
nonlinear laws and interactions requires the development of new qual
itativetechniques for analyzing the motions in nonlinear dynamical
systems
What is in this b o ok
This b o ok intro duces qualitative bifurcation theory symb olic dynam
ics etc and quantitative p erturbation theory numerical metho ds
etc metho ds that can b e used in analyzing a nonlinear dynamical
system Further it provides a basic set of exp erimental techniques re
quired to set up and observe nonlinear phenomena in the lab oratory
V I Arnold and A Avez Ergodic problems of classical mechanics W A
Benjamin New York
What is in this b o ok
Some of these metho ds go backtoPoincar e s original work in the last
century while many others such as computer simulations are more
recent A wide assortment of seemingly disparate techniques is used
in the analysis of the humblest nonlinear dynamical system Whereas
linear theory resembles an edi ce built up on the principle of sup erp osi
tion nonlinear theory more closely resembles a to olb ox in whichmany
of the essential to ols have b een b orrowed from the lab oratories of many
di erent friends
To paraphrase Tolstoy all linear systems resemble one another but
each nonlinear system is nonlinear in its own way Therefore on our
rst encounter with a new nonlinear system we need to search our to ol
box for the prop er diagnostic to ols p ower sp ectra fractal dimensions
p erio dic orbit extraction etc so that we can identify and characterize
the nonlinear and chaotic structures And next we need to analyze and
unfold these structures with the help of additional to ols and metho ds
computer simulations simpli ed geometric mo dels universality the
ory etc to nd those prop erties that are common to a large class of
nonlinear systems
The to ols in our to olb ox are collected from scientists in a wide
range of disciplines mathematics physics computing engineering
economics and so on Each discipline has develop ed a di erentdi
alect and sometimes even a new language in which to discuss nonlinear
problems And so one challenge facing a new researcher in nonlinear
dynamics is to develop some uency in these di erent dialects
It is typical in many elds from cabinet making to mathematics to
intro duce the tyro rst to the tedious elements and next when these
basic elements are mastered to intro duce her to the joys of the celestial
whole The cabinet maker rst learns to sweep and sand and measure
and hold Likewise the aspiring mathematician learns how to express
limits take derivatives calculate integrals and make substitutions
All to o often the consequence of an intro duction through tedium is
the destruction of the inquisitive eager spirit of inquiry Wehopeto
diminish this tedium bytying the eagerness of the student to pro jects
and exp eriments that illustrate nonlinear concepts In a few words
wewant the student to get her hands dirty Then with maturityand
insight b orn from rsthand exp erience she will b e ready to ll in the
big picture with rigorous de nitions and more comprehensive study
Intro duction
The study of nonlinear dynamics is eclectic selecting what app ears
to b e most useful among various and diverse theories and metho ds This
p oses an additional challenge since the skills required for researchin
nonlinear dynamics can range from a knowledge of some sophisticated
mathematics hyp erb olicity theory to a detailed understanding of the
nuts and b olts of computer hardware binary arithmetic digitizers
Nonlinear dynamics is not a tidy sub ject but it is vital
The common thread throughout all nonlinear dynamics is the need
and desire to solve nonlinear problems byhookorby cro ok In
deed many to ols in the nonlinear dynamicist s to olb ox originally were
crafted as the solution to a sp eci c exp erimental problem or applica
tion Only after solving many individual nonlinear problems did the
common threads and structures slowly emerge
The rst half of this b o ok Chapters and uses a similar exp er
imental approach to nonlinear dynamics and is suitable for an advanced
undergraduate course Our approach seeks to develop and motivate the
study of nonlinear dynamics through the detailed analysis of a few sp e
ci c systems that can b e realized by desktop exp eriments the p erio d
doubling route to chaos in a b ouncing ball the symb olic analysis of
the quadratic map and the quasip erio dic and chaotic vibrations of a
string The detailed analysis of these examples develops intuition for
and motivates the study of nonlinear systems In addition analysis
and simulation of these elementary examples provide ample practice
with the to ols in our nonlinear dynamics to olb ox The second half of
the b o ok Chapters and provides a more formal treatmentofthe
theory illustrated in the desktop exp eriments thereby setting the stage
for an advanced or graduate level course in nonlinear dynamics In
addition Chapters and provide the more advanced student or re
searcher with a concise intro duction to the mathematical foundations
of nonlinear dynamics as well as intro ducing the top ological approach
toward the analysis of chaos
The p edagogical approach also di ers b etween the rst and second
half of the b o ok The rst half tends to intro duce new concepts and
vo cabulary through usage example and rep eated exp osure We b elieve
this metho d is p edagogically sound for a rst course and is reminiscent
of teaching metho ds found in an intensive foreign language course In
the second half of the b o ok examples tend to follow formal de nitions
Some Terminology
as is more common in a traditional mathematics course
Although a linear reading of the rst three chapters of this text is
the most useful it is also p ossible to pickandcho ose material from dif
ferent sections to suit sp eci c course needs A mixture of mathematical
theoretical exp erimental and computational metho ds are employed in
the rst three chapters The following table provides a rough road map
to the typ e of material found in each section
Mathematical Theoretical Exp erimental Computational
M athematica usage
App endix B App endices
A C D E F G
The mathematical sections present the core material for a mathe
matical dynamical systems course The exp erimental sections present
the core exp erimental techniques used for lab oratory work with a low
dimensional chaotic system For instance those interested in exp eri
mental techniques could turn directly to the exp erimental sections for
the information they seek
Some Terminology
Maps Flows and Fractals
In this section we heuristically intro duce some of the basic terminology
used in nonlinear dynamics This material should b e read quickly
as background to the rest of the b o ok It might also b e helpful to
read App endix H Historical Comments b efore delving into the more
technical material For precise de nitions of the mathematical notions
intro duced in this section we highly recommend V I Arnold s masterful
intro duction to the theory of ordinary di erential equations The goal
in this section is to b egin using the vo cabulary of nonlinear dynamics
even b efore this vo cabulary is precisely de ned
V I Arnold Ordinary di erential equations MIT Press Cambridge MA
Also see D K Arrowsmith and C M Place Ordinary di erential equations
Chapman and Hall New York
Intro duction
Figure a The surfaces of the three dimensional ob jects are two
dimensional manifolds b Examples of ob jects that are not manifolds
Flows and Maps
A geometric formulation of the theory of di erential equations says that
a di erential equation is a vector eld on a manifold To understand
this de nition we present an informal description of a manifold and a
vector eld
A manifold is any smo oth geometric space line surface solid The
smo othness condition ensures that the manifold cannot haveany sharp
edges An example of a one dimensional manifold is an in nite straight
line A di erent one dimensional manifold is a circle Examples of
two dimensional manifolds are the surface of an in nite cylinder the
surface of a sphere the surface of a torus and the unb ounded real plane
Fig Three dimensional manifolds are harder to visualize The
simplest example of a three dimensional manifold is unb ounded three
space R The surface of a cone is an example of a two dimensional
surface that is not a manifold At the ap ex of the cone is a sharp p oint
which violates the smo othness condition for a manifold Manifolds are
useful geometric ob jects b ecause the smo othness condition ensures that
a lo cal co ordinate system can b e erected at eachandevery p ointon
the manifold
A vector eld is a rule that smo othly assigns a vector a directed
Some Terminology
line segment to eachpoint of a manifold This rule is often written
as a system of rst order di erential equations To see how this works
consider again the linear di erential equation
d x
x
dt
Let us rewrite this second order di erential equation as a system of
two rst order di erential equations byintro ducing the new variable v
velo city de ned by dx dt v sothat
dx
v
dt
dv
x
dt
The manifold in this example is the real plane R which consists of
the ordered pair of variables x v Each p oint in this plane repre
sents an individual state or p ossible initial condition of the system
And the collection of all p ossible states is called the phase space of the
system A pro cess is said to b e deterministic if b oth its future and
past states are uniquely determined by its present state A pro cess is
called semideterministic when only the future state but not the past
is uniquely determined by the present state Not all physical systems
are deterministic as the b ouncing ball system which is only semide
terministic of Chapter demonstrates Nevertheless full determinism
is commonly assumed in the classical scienti c world view
A system of rst order di erential equations assigns to each p oint
of the manifold a vector thereby forming a vector eld on the manifold
Fig In our example eachpoint of the phase plane x v gets
assigned a vector v x which forms rings of arrows ab out the origin
Fig A solution to a di erential equation is called a trajectory
or an integral curve since it results from integrating the di erential
equations of motion An individual vector in the vector eld deter
mines how the solution b ehaves lo cally It tells the tra jectory to go
thataway The collection of all solutions or integral curves is called
the ow Fig
When analyzing a system of di erential equations it is imp ortant
to present b oth the equations and the manifold on which the equations
Intro duction
Figure Examples of vector elds on di erent manifolds
Figure Vector eld and ow for a linear di erential equation
are sp eci ed It is often p ossible to simplify our analysis by transferring
the vector eld to a di erent manifold therebychanging the top ology
of the phase space see section Topology is a kind of geometry
which studies those prop erties of a space that are unchanged under
areversible continuous transformation It is sometimes called rubb er
sheet geometry A basketball and a fo otball are identical to a top ologist
They are b oth top ological spheres However a torus and a sphere
are di erent top ological spaces as you cannot push or pull a sphere into
a torus without rst cutting up the sphere Top ology is also de ned
as the study of closeness within neighb orho o ds Top ological spaces
can b e analyzed by studying whichpoints are close to or in the
neighborhood of other points Consider the line segmentbetween
Some Terminology
Figure Typical motions in a planar vector eld a source b
sink c saddle and d limit cycle
and The endp oints and are far away they aren t neighb ors But
if we glue the ends together to form a circle then the endp oints b ecome
identical and the p oints around and have a new set of neighb ors
In its grandest form Poincar e s program to study the qualitativebe
havior of ordinary di erential equations would require us to analyze the
generic dynamics of all vector elds on all manifolds We are nowhere
near achieving this goal yet Poincar ewas inspired to carry out this
program by his success with the Swedish mathematician Ivar Bendixson
in analyzing all typical b ehavior for di erential equations in the plane
As illustrated in Figure the Poincar e Bendixson Theorem says that
typically no more than four kinds of motion are found in a planar vector
eld those of a source sink sadd le andlimit cycle In particular no
chaotic motion is possible in time independent planar vector elds To
get chaotic motion in a system of di erential equations one needs three
dimensions that is a vector eld on a three dimensional manifold
Intro duction
The asymptotic motions t limit sets of a ow are character
ized by four general typ es of b ehavior In order of increasing complexity
these are equilibrium points periodic solutions quasiperiodic solutions
and chaos
An equilibrium point of a owisaconstant time indep endent so
lution The equilibrium solutions are lo cated where the vector eld
vanishes The source in Figure a is an example of an unstable
equilibrium solution Tra jectories near to the source moveaway from
the source as time go es by The sink in Figure b is an example of
a stable equilibrium solution Tra jectories near the sink tend toward it
as time go es by
A periodic solution of a ow is a time dep endent tra jectory that
precisely returns to itself in a time T called the period A p erio dic
tra jectory is a closed curve Like an equilibrium p oint a p erio dic tra
jectory can b e stable or unstable dep ending on whether nearby tra jec
tories tend toward or away from the p erio dic cycle One illustration of
a stable p erio dic tra jectory is the limit cycle shown in Figure d
A quasiperiodic solution is one formed from the sum of p erio dic solu
tions with incommensurate p erio ds Two p erio ds are incommensurate
if their ratio is irrational The ability to create and control p erio dic
and quasip erio dic cycles is essential to mo dern so ciety clo cks elec
tronic oscillators pacemakers and so on
An asymptotic motion that is not an equilibrium p oint p erio dic
or quasip erio dic is often called chaotic Thiscatchall use of the term
chaos is not very sp eci c but it is practical Additionally we require
that a chaotic motion is a b ounded asymptotic solution that p ossesses
sensitive dep endence on initial conditions two tra jectories that b egin
arbitrarily close to one another on the chaotic limit set start to diverge
so quickly that they b ecome for all practical purp oses uncorrelated
Simply put a chaotic system is a deterministic system that exhibits
random uncorrelated b ehavior This apparent random b ehavior in
a deterministic system is illustrated in the b ouncing ball system see
section A more rigorous de nition of chaos is presented in section
All of the stable asymptotic motions or limit sets just describ ed
e g sinks stable limit cycles are examples of attractors The unsta
ble limit sets e g sources are examples of repel lers The term strange
Some Terminology
attractor strange rep eller is used to describ e attracting rep elling
limit sets that are chaotic We will get our rst lo ok at a strange at
tractor in a physical system when we study the b ouncing ball system
in Chapter
Maps are the discrete time analogs of ows While ows are sp eci ed
by di erential equations maps are sp eci ed by di erence equations
A p oint on a tra jectory of a ow is indicated by a real parameter t
whichwe think of as the time Similarly a pointintheorbitofa
map is indexed byaninteger subscript n whichwe think of as the
discrete analog of time Maps and ows will b e the two primary typ es
of dynamical systems studied in this b o ok
Maps di erence equations are easier to solvenumerically than
ows di erential equations Therefore many of the earliest numerical
studies of chaos b egan by studying maps A famous map exhibiting
chaos studied bytheFrench astronomer Michel H enon now
known as the H enon map is
y x x
n n
n
y x
n n
where n is an integer index for this pair of nonlinear coupled di er
ence equations with and b eing the parameter values
most commonly studied The H enon map carries a p ointintheplane
x y to some new p oint x y An orbit of a map is the sequence
of p oints generated by some initial condition of a map For instance if
we start the H enon map at the p oint x y we nd that
the orbit for this pair of initial conditions is
x
y
x
y
and so on to generate x y x y etc Unlike planar di eren
tial equations this two dimensional di erence equation can generate
chaotic orbits In fact in Chapter we will study a one dimensional
Intro duction
Figure A Poincar e map for a three dimensional ow with a two
dimensional cross section
di erence equation called the quadratic map which can also generate
chaotic orbits
The H enon map is an example of a di eomorphism of a manifold in
this case the manifold is the plane R A map is a homeomorphism if
it is bijective one to one and onto continuous and has a continuous
inverse A di eomorphism is a di erentiable homeomorphism A map
with an inverse is called invertible A map without an inverse is called
noninvertible
Maps exhibit similar typ es of asymptotic b ehavior as ows equi
librium p oints p erio dic orbits quasip erio dic orbits and chaotic orbits
There are many similarities and a few imp ortant di erences b etween
the theory and language describing the dynamics of maps and ows
For a detailed comparison of these two theories see Arrowsmith and
Place Anintroduction to dynamical systems
The dynamics of ows and maps are closely related The study of
a ow can often b e replaced by the study of a map One prescription
for doing this is the so called Poincar emapofa ow As illustrated
in Figure a cross section of the ow is obtained bycho osing some
surface transverse to the ow A cross section for a three dimensional
Some Terminology
ow is shown in the illustration and is obtained bycho osing the x y
plane x y z The ow de nes a map of this cross section to itself
and this map is an example of a Poincar emap also called a rst return
map A tra jectory of the ow carries a p oint x t y t into a new
p oint x t y t And this in turn go es to the p oint x t y t
In this way the ow generates a map of a p ortion of the plane and an
orbit of this map consists of the sequence of p oints
x y x t y t z z
x y x t y t z z
x y x t y t z z
and so on
There is another reason for studying maps To quote Steve Smale
on the di eomorphism problem
T here is a second and more imp ortant reason for study
ing the di eomorphism problem b esides its great natural
beauty That is the same phenomena and problems of
the qualitative theory of ordinary di erential equations are
present in their simplest form in the di eomorphism prob
lem Having rst found theorems in the di eomorphism
case it is usually a secondary task to translate the results
backinto the di erential equations framework
The rst dynamical system we will study the b ouncing ball system
illustrates more fully the close connection b etween maps and ows
Binary Arithmetic
Before turning to nonlinear dynamics prop er we need some familiarity
with the binary numb er system Consider the problem of converting a
fraction b etween and x written in decimal base to a
binary numb er base The formal expansion for a binary fraction in
S Smale Di erential dynamical systems Bull Am Math So c
Intro duction
powers of is
decimal x
binary
where f g The goal is to nd the s for a given decimal
i i
fraction For example if x then
x
binary
The general pro cedure for converting a decimal fraction less than
one to binary is based on rep eated doublings in which the ones or
carry digit is used for the s This is illustrated in the following
i
calculation for x
so
x binary
Fractals
Nature ab ounds with intricate fragmented shap es and structures in
cluding coastlines clouds lightning b olts and snow akes In
Benoit Mandelbrot coined the term fractal to describ e such irregular
shap es The essential feature of a fractal is the existence of a similar
Some Terminology
Figure Construction of Cantor s middle thirds set
structure at all length scales That is a fractal ob ject has the prop erty
that a small part resembles a larger part which in turn resembles the
whole ob ject Technically this prop erty is called self similarity and is
theoretically describ ed in terms of a scaling relation
Chaotic dynamical systems almost inevitably give rise to fractals
And fractal analysis is often useful in describing the geometric structure
of a chaotic dynamical system In particular fractal ob jects can b e
assigned one or more fractal dimensions which are often fractional
that is they are not integer dimensions
Toseehowthisworks consider a Cantor set which is de ned re
cursively as follows Fig At the zeroth level the construction of
the Cantor set b egins with the unit interval that is all p oints on the
line b etween and The rst level is obtained from the zeroth level
by deleting all p oints that lie in the middle third that is all p oints
between and The second level is obtained from the rst level
by deleting the middle third of eachinterval at the rst level that is
all p oints from to and to In general the next level
is obtained from the previous level by deleting the middle third of all
intervals at the previous level This pro cess continues forever and the
result is a collection of p oints that are tenuously cut out from the unit
n
interval At the nth level the set consists of segments each of which
Intro duction
n
has length l so that the length of the Cantor set is
n
n
n
lim
n
In the s the mathematician Hausdor develop ed another way
to measure the size of a set He suggested that we should examine
the numb er of small intervals N needed to cover the set at a scale
The measure of the set is calculated from
d
f
lim N
An example of a fractal dimension is obtained byinverting this equa
tion
ln N
A
d lim
f
ln
Returning to the Cantor set we see that at the nth level the length of
n
and the number of intervals needed the covering intervals are
n
to cover all segments at the nth level is N Taking the limits
n we nd
n
ln N ln ln
A
d lim lim
f
n
n
ln ln
ln
The middle thirds Cantor set has a simple scaling relation b ecause
the factor is all that go es into determining the successive levels A
further elementary discussion of the middle thirds Cantor set is found
in Devaney s Chaos fractals and dynamics In general fractals arising
in a chaotic dynamical system have a far more complex scaling relation
usually involving a range of scales that can dep end on their lo cation
within the set Such fractals are called multifractals
References and Notes
References and Notes
Some p opular accounts of the history and folklore of nonlinear dynam
ics and chaos include
A Fisher Chaos The ultimate asymmetry MOSAIC pp Jan
uary February
J P Crutch eld J D Farmer N H Packard and R S Shaw Chaos Sci Am
pp
J Gleick Chaos Making a new science Viking New York
I Stewart Does god play dice The mathematics of chaos Basil Blackwell Cam
bridge MA
The following b o oks are helpful references for some of the material cov
ered in this b o ok
R Abraham and C Shaw Dynamics The geometry of behavior Vol Aerial
Press Santa Cruz CA
D K Arrowsmith and C M Place An introduction to dynamical systems Cam
bridge University Press New York
G L Baker and J P Gollub Chaotic dynamics Cambridge University Press New
York
P Berg e Y Pomeau and C Vidal Order within chaos John Wiley New York
R L Devaney Anintroduction to chaotic dynamical systems second ed
Addison Wesley New York
E A Jackson Perspectives of nonlinear dynamics Vol Cambridge University
Press New York
F Mo on Chaotic vibrations John Wiley New York
J Thompson and H Stewart Nonlinear dynamics and chaos John Wiley New
York
S Rasband Chaotic dynamics of nonlinear systems John Wiley New York
Intro duction
S Wiggins Introduction to applied nonlinear dynamical systems and chaos Springer
Verlag New York
These review articles provide a quickintro duction to the current re
search problems and metho ds of nonlinear dynamics
J P Eckmann Roads to turbulence in dissipative dynamical systems Rev Mo d
Phys pp
J P Eckmann and D Ruelle Ergo dic theory of chaos and strange attractors Rev
Mo d Phys pp
C Greb ogi E Ott and J Yorke Chaos strange attractors and fractal basin
b oundaries in nonlinear dynamics Science pp
E Ott Strange attractors and chaotic motions of dynamical systems Rev Mo d
Phys pp
T Parker and L Chua Chaos A tutorial for engineers Pro c IEEE pp
R Shaw Strange attractors chaotic b ehavior and information ow Z Naturforsch
a pp
Advanced theoretical results are describ ed in the following b o oks
V I Arnold Geometrical methods in the theory of ordinary di erential equations
second ed Springer Verlag New York
J Guckenheimer and P Holmes Nonlinear oscil lations dynamical systems and
bifurcations of vector elds second printing Springer Verlag New York
D Ruelle Elements of di erentiable dynamics and bifurcation theory Academic
Press New York
S Wiggins Global bifurcations and chaos Springer Verlag New York
Chapter
Bouncing Ball
Intro duction
Consider the motion of a ball b ouncing on a p erio dically vibrating ta
ble The bouncing bal l system is illustrated in Figure and arises
quite naturally as a mo del problem in several engineering applications
Examples include the generation and control of noise in machinery such
as jackhammers the transp ortation and separation of granular solids
such as rice and the transp ortation of comp onents in automatic assem
bly devices which commonly employ oscillating tracks These vibrating
tracks are used to transp ort parts muchlikea conveyor b elt
Assume that the ball s motion is con ned to the vertical direction
and that b etween impacts the ball s height is determined by Newton s
laws for the motion of a particle in a constantgravitational eld A
nonlinear force is applied to the ball when it hits the table At impact
the ball s velo city suddenly reverses from the downward to the upward
direction Fig
The b ouncing ball system is easy to study exp erimentally One
exp erimental realization of the system consists of little more than a ball
b earing and a p erio dically driven loudsp eaker with a concave optical
lens attached to its surface The ball b earing will rattle on top of
this lens when the sp eaker s vibration amplitude is large enough The
curvature of the lens is chosen so as to help fo cus the ball s motion in
the vertical direction
CHAPTER BOUNCING BALL
Figure Ball b ouncing on an oscillating table
Impacts b etween the ball and lens can b e detected by listening to
the rhythmic clicking patterns pro duced when the ball hits the lens A
piezo electric lm which generates a small currentevery time a stress
is applied is fastened to the lens and acts as an impact detector The
piezo electric lm generates a voltage spike at each impact This spike
is monitored on an oscilloscop e thus providing a visual representation
of the ball s motion Aschematic of the b ouncing ball machine is
shown in Figure More details ab out its construction are provided
in reference
The ball s motion can b e describ ed in several equivalentways The
simplest representation is to plot the ball s height and the table s height
measured from the ground as a function of time Between impacts the
graph of the ball s vertical displacementfollows a parab olic tra jectory
as illustrated in Figure a The table s vertical displacementvaries
sinusoidally If the ball s height is recorded at discrete time steps
fx t x t x t x t g
i n
then wehavea time series of the ball s height where x t is the height
i
of the ball at time t
i
Another view of the ball s motion is obtained by plotting the ball s
heightonthevertical axis and the ball s velo city on the horizontal axis
The plot shown in Figure b is essentially a phase space represen
tation of the ball s motion Since the ball s height is b ounded so is
the ball s velo city Thus the phase space picture gives us a description
INTRODUCTION
Figure Schematic for a b ouncing ball machine
of the ball s motion that is more compact than that given by a plot of
the time series Additionally the sudden reversal in the ball s velo city
at impact from p ositive to negative is easy to see at the b ottom of
Figure b Between impacts the graph again follows a parab olic
tra jectory
Yet another representation of the ball s motion is a plot of the ball s
velo city and the table s forcing phase at each impact This is the so
called impact map and is shown in Figure c The impact map go es
to a single p oint for the simple p erio dic tra jectory shown in Figure
The vertical co ordinate of this p oint is the ball s velo city at impact and
the horizontal co ordinate is the table s forcing phase This phase
is de ned as the pro duct of the table s angular frequency and the
time t
t T
where T is the forcing p erio d Since the table s motion is p erio dic
in the phase variable we usually consider the phase mo d which
means we divide by and take the remainder
mo d remainder
CHAPTER BOUNCING BALL
Figure Simple p erio dic orbit of a b ouncing ball a height vs time
b phase space height vs velo city c impact map velo cityand
forcing phase at impact Generated by the Bouncing Ball program
A time series phase space and impact map plot are presented to
gether in Figure for a complex motion in the b ouncing ball system
This particular motion is an example of a nonp erio dic orbit known as
a strange attractor The impact map Figure c is a compact and
abstract representation of the motion In this particular example we
see that the ball never settles down to a p erio dic motion in whichit
would impact at only a few p oints but rather explores a wide range
of phases and velo cities We will saymuch more ab out these strange
tra jectories throughout this b o ok but rightnowwe turn to the details of mo deling the dynamics of a b ouncing ball
MODEL
Figure Strange orbit of a b ouncing ball a height vs time
b phase space c impact map Generated by the Bouncing Ball
program
Mo del
To mo del the b ouncing ball system we assume that the table s mass
is much greater than the ball s mass and the impact b etween the ball
and the table is instantaneous These assumptions are realistic for the
exp erimental system describ ed in the previous section and simply mean
that the table s motion is not a ected by the collisions The collisions
are usually inelastic that is a little energy is lost at each impact If no
energy is lost then the collisions are called elastic We will examine b oth
cases in this b o ok the case in which energy is dissipated dissipative
and the case in which energy is conserved conservative
CHAPTER BOUNCING BALL
Stationary Table
First though wemust gure out how the ball s velo citychanges at
each impact Consider two di erent reference frames the ball s motion
as seen from the ground the ground s reference frame and the ball s
motion as seen from the table the table s reference frame Begin by
considering the simple case where the table is stationary and the two
reference frames are identical As wewillshow shortly understanding
the stationary case will solve the nonstationary case
Let v b e the ball s velo city right b efore the k th impact and let
k
v b e the ball s velo cityright after the k th impact The prime nota
k
tion indicates a velo cityimmediately b efore an impact If the table
the ball is stationary and the collisions are elastic then v v
k
k
reverses direction but do es not change sp eed since there is no energy
loss If the collisions are inelastic and the table is stationary then the
ball s sp eed will b e reduced after the collision b ecause energy is lost
v v where is the coe cient of restitution The
k
k
constant is a measure of the energy loss at each impact If
the system is conservative and the collisions are elastic The co e cient
of restitution is strictly less than one for inelastic collisions
Impact Relation for the Oscillating Table
When the table is in motion the ball s velo cityimmediately after an
impact will have an additional term due to the kick from the table To
calculate the change in the ball s velo city imagine the motion of the
ball from the table s p ersp ective The key observation is that in the
table s reference frame the table is always stationary The ball however
appearstohave an additional velo citywhich is equal to the opp osite
of the table s velo city in the ground s reference frame Therefore to
calculate the ball s change in velo citywe can calculate the change in
velo city in the table s reference frame and then add the table s velo city
to get the ball s velo city in the ground s reference frame In Figure
weshow the motion of the ball and the table in b oth the ground s and
the table s reference frames
The co e cient of restitution is called the damping coe cient in the Bouncing Ball program
MODEL
Figure Motion of the ball in the reference frame of the ground a
and the table b
Let u b e the table s velo city in the ground s reference frame Fur
k
andv be the velo city in the table s reference frame imme ther letv
k
k
diately b efore and after the k th impact resp ectively The bar denotes
measurements in the table s reference frame the unbarred co ordinates
are measurements in the ground s reference frame Then in the table s
reference frame
v v
k
k
since the table is always stationary To nd the ball s velo city in the
ground s reference frame wemust add the table s velo city to the ball s
apparentvelo city
v v u v v u
k k k k
k k
or equivalently
v v u v v u
k k k k
k k
Therefore in the ground s reference frame equation b ecomes
v u v u
k k k
k
when it is rewritten using equation Rewriting equation
gives the velo city v after the k th impact as
k
v u v
k k
k
This last equation is known as the impact relation It says the kick
from the table contributes u to the ball s velo city k
CHAPTER BOUNCING BALL
The Equations of Motion Phase
and Velo city Maps
To determine the motion of the ball wemust calculate the times hence
phases from eq when the ball and the table collide An impact
o ccurs when the di erence b etween the ball p osition and the table p osi
tion is zero Between impacts the ball go es up and down according to
Newton s law for the motion of a pro jectile in a constantgravitational
eld of strength g Since the motion b etween impacts is simple we will
present the motion of the ball in terms of an impact map that is some
rule that takes as input the currentvalues of the impact phase and
impact velo city and then generates the next impact phase and impact
velo city
Let
g t t x t x v t t
k k k k
b e the ball s p osition at time t after the k th impact where x is the
k
p osition at the k th impact and t is the time of the k th impact and let
k
s t A sin t
b e the table s p osition with an amplitude A angular frequency and
phase at t We add one to the sine function to ensure that the
table s amplitude is always p ositive The di erence in p osition b etween
the ball and table is
d t x t s t
which should always b e a non negative function since the ball is never
b elow the table The rst value at which d t t t implicitly
k
de nes the time of the next impact Substituting equations and
into equation and setting d t to zero yields
x v t t g t t A sin t
k k k k k k k
For a discussion of the motion of a particle in a constantgravitational eld see
anyintro ductory physics text suchasR Weidner and R Sells Elementary Physics
Vol Allyn and Bacon Boston pp
MODEL
Equation can b e rewritten in terms of the phase when the iden
ti cation t is made b etween the phase variable and the time
variable This leads to the implicit phase map of the form
A sin v
k k k k
A sin g
k k k
where is the next for which d In deriving equation
k
we used the fact that the table p osition and the ball p osition are
identical at an impact that is x A sin
k k
An explicit velocity map is derived directly from the impact relation
equation as
v A cos t v g t t
k k k k k
or in the phase variable
v A cos v g
k k k k k
noting that the table s velo city is just the time derivative of the table s
p osition u t s t ds dt A cos t and that b etween
impacts the ball is sub ject to the acceleration of gravity so its velo city
is given by v g t t The overdot is Newton s original notation
k k
denoting di erentiation with resp ect to time
The implicit phase map eq and the explicit velo citymap
eq constitute the exact mo del for the b ouncing ball system
The dynamics of the b ouncing ball are easy to simulate on a computer
using these two equations Unfortunately the phase map is an implicit
algebraic equation for the variable that is cannot b e iso
k k
lated from the other variables Tosolve the phase function for a
k
numerical algorithm is needed to lo cate the zeros of the phase func
tion see App endix A Still this presents little problem for numerical
simulations or even as we shall see for a go o d deal of analytical work
Parameters
The parameters for the b ouncing ball system should b e determined
b efore we continue our analysis The relevant parameters with typical
CHAPTER BOUNCING BALL
Parameter Symbol Exp erimental values
Co e cient of restitution
Table s amplitude A cm
Table s p erio d T s
Gravitational acceleration g cm s
Frequency f f T
Angular frequency f
A g Normalized acceleration
Table Reference values for the Bouncing Ball System
exp erimental values are listed in Table In an exp erimental system
the table s frequency or the table s amplitude of oscillation is easy to
adjust with the function generator The co e cient of restitution can
also b e varied by using balls comp osed of di erent materials Steel
balls for instance are relatively hard and have a high co e cientof
restitution Balls made from brass glass plastic or wo o d are softer
and tend to dissipate more energy at impact
As wewillshow in the next section the physical parameters listed in
Table are related By rescaling the variables it is p ossible to show
that there are only two fundamental parameters in this mo del For
our purp oses we will take these to b e the co e cient of restitution
and a new parameter which is essentially prop ortional to A The
parameter is in essence a normalized acceleration and it measures
the violence with which the table oscillates up and down
High Bounce Approximation
In the high b ounce approximation we imagine that the table s displace
ment amplitude is always small compared to the ball s maximum height
This approximation is depicted in Figure where the ball s tra jectory
is p erfectly symmetric about its midpoint and therefore
v v
k
k
HIGH BOUNCE APPROXIMATION
Figure Symmetric orbit in the high b ounce approximation
The velo city of the ball b etween the k th and k st impacts is given
by
v t v g t t
k k
Atthe k st impact the velo cityis v and the time is t so
k
k
v g t t v
k k k
k
Using equation and simplifying weget
v t t
k k k
g
whichisthetime map in the high b ounce approximation
To ndthevelo city map in this approximation we b egin with the
impact relation eq
v u v
k k
k
u v
k k
where the last equality follows from the high b ounce approximation
equation The table s velo cityatthe k st impact can b e
written as
u A cos t
k k
A cos t v g
k k
when the time map equation is used Equations and
give the velo city map in the high b ounce approximation
v v A cos t v g
k k k k
CHAPTER BOUNCING BALL
The impact equations can b e simpli ed somewhat bychanging to
the dimensionless quantities
t
v g and
A g
which recasts the time map eq and the velo city map eq
into the explicit mapping form
k k k
f f
cos
k k k k
In the sp ecial case where this system of equations is known as the
standard map The subscripts of f explicitly show the dep endence
of the map on the parameters and The mapping equation
is easy to solve on a computer Given an initial condition the
map explicitly generates the next impact phase and impact velo cityas
f and this in turn generates f f
f f etc where in this notation the sup erscript n in
n
f indicates functional comp osition see section Unlike the exact
mo del b oth the phase map and the velo city map are explicit equations
in the high b ounce approximation
The high b ounce approximation shares many of the same qualita
tive prop erties of the exact mo del for the b ouncing ball system and it
will serve as the starting p oint for several analytic calculations How
ever for comparisons with exp erimental data it is worthwhile to put
the extra e ort into numerically solving the exact equations b ecause
the high b ounce mo del fails in at least two ma jor ways to mo del the
actual physical system First the high b ounce mo del can generate
solutions that cannot p ossibly o ccur in the real system These unphys
ical solutions o ccur for very small b ounces at negative table velo cities
where it is p ossible for the ball to b e pro jected downward b eneath the
table That is the ball can pass through the table in this approxi
mation Second this approximation cannot repro duce a large class of
real solutions called sticking solutions which are discussed in sec
tion Fundamentally this is b ecause the map in the high b ounce
QUALITATIVE DESCRIPTION OF MOTIONS
approximation is invertible whereas the exact mo del is not invertible
In the exact mo del there exist some solutions in particular the stick
ing solutions for whichtwo or more orbits are mapp ed to the same
identical p oint Thus the map at this p oint do es not have a unique
inverse
Qualitative Description of Motions
In sp ecifying an individual solution to the b ouncing ball system we
need to know b oth the initial condition that is the initial impact
phase and impact velo city of the ball v and the relevant system
parameters A and T Then to nd an individual tra jectory allwe
need to do is iterate the mapping for the appropriate mo del However
nding the solution for a single tra jectory gives little insightinto the
global dynamics of the system As stressed in the Intro duction we
are not interested so much in solving an individual orbit but rather
in understanding the b ehavior of a large collection of orbits and when
p ossible the system as a whole
An individual solution can b e represented by a curve in phase space
In considering a collection of solutions we will need to understand the
behavior not of a single curve in phase space but rather of a bundle of
adjacent curves a region in phase space Similarly in the impact map
wewant to consider a collection of initial conditions a region in the
impact map In general the future of an orbit is well de ned bya ow
or mapping The fate of a region in the phase space or the impact map
is de ned by the collective futures of each of the individual curves or
p oints resp ectively in the region as is illustrated in Figure
Anumb er of questions can and will b e asked ab out the evolution
of a region in phase space or in the impact map Do regions in the
phase space expand or contract as the system evolves In the b ounc
ing ball system a bundle of initial conditions will generally contract in
area whenever the system is dissipative a little energy is lost at each
impact and this results in a shrinkage of our initial region or patch in
phase space see section for details Since this region is shrink
ing this raises many questions that will b e addressed throughout this
b o ok such as where do all the orbits go howmuch of the initial area
CHAPTER BOUNCING BALL
Figure a Evolution of a region in phase space b Recurrent
regions in the phase space of a nonlinear system
remains and what do the orbits do on this remaining set once they get
to wherever they re going This turns out to b e a subtle collection of
questions For instance even the question of what we mean by area
gets tricky b ecause there is more than one useful notion of the area
or measure of a set Another related question is do these regions in
tersect with themselvesastheyevolve see Figure The answer
is generally yes they do intersect and this observation will lead us to
study the rich collection of recurrence structures of a nonlinear system
A simple question wecananswer is do es there exist a closed
simply connected subset or region of the whole phase space or impact
map such that all the orbits outside this subset eventually enter into it
and once inside they never get out again If such a subset exists it is
called a trapping region Establishing the existence of a trapping region
can simplify our general problem somewhat b ecause instead of consid
ering all p ossible initial conditions we need only consider those initial
conditions inside the trapping region since all other initial conditions
will eventually end up there
Trapping Region
To nd a trapping region for the b ouncing ball system we will rst nd
an upp er b ound for the next outgoing velo city v by lo oking at the
k
QUALITATIVE DESCRIPTION OF MOTIONS
previous value v We will then nd a lower b ound for v These
k k
b ounds give us the b oundaries for a trapping region in the b ouncing
ball s impact map v which imply a trapping region in phase space
i i
To b ound the outgoing velo city we b egin with equation in
the form
v v A cos t g t t
k k k k k
The rst term on the right hand side is easy to b ound To b ound the
second term we rst lo ok at the average ball velo citybetween impacts
whichisgiven by
g t t v v
k k k k
Rearranging this expression gives
t t v v
k k k k
g
Equation now b ecomes
v v A cos t v v
k k k k k
Noting that the average table velo citybetween impacts is the same as
the average ball velo citybetween impacts see Prob we ndthat
v v A
k k
If we de ne
v A
max
and let v v then v v v or
k max k k k
v v
k k
In this case it is essential that the system b e dissipative for a
trapping region to exist In the conservative limit no trapping region
exists it is p ossible for the ball to reach in nite heights and velo cities when no energy is lost at impact
CHAPTER BOUNCING BALL
To nd a lower b ound for v we simply realize that the velo city
k
after impact must always b e at least that of the table
v A v
k min
For the b ouncing ball system the compact trapping region D given
by
D f v j v v v g
min max
is simply a strip b ounded by v and v Toprove that D is a
min max
trapping region we also need to showthatv cannot approach v
max
asymptotically and that once inside D the orbit cannot leave D these
calculations are left to the reader see Prob The previous cal
culations show that all orbits of the dissipative b ouncing ball system
will eventually enter the region D and b e trapp ed there
Equili bri um Solutions
Once the orbits enter the trapping region where do they go next To
answer this question we rstsolve for the motion of a ball b ouncing on
a stationary table Then we will imagine slowly turning up the table
amplitude
If the table is stationary then the high b ounce approximation is
no longer approximate but exact Setting A inthevelo city map
equation immediatelygives
v v
k k
Using the time map t t g v the co e cient of restitution is
k k k
easy to measure by recording three consecutive impact times since
t t
k k
t t
k k
To ndhow long it takes the ball to stop b ouncing consider the sum
of the di erences of consecutive impact times
X
t t
n k k k
n
QUALITATIVE DESCRIPTION OF MOTIONS
Since is the summation of a geometric series
k k
X
n
n
which can b e summed for After an in nite numb er of b ounces
the ball will come to a halt in a nite time
For these equilibrium solutions all the orbits in the trapping region
come to rest on the table When the table s acceleration is small the
picture do es not change much The ball comes to rest on the oscillating
table and then moves in unison with the table from then on
Sticking Solutions
Now that the ball is moving with the table what happ ens as weslowly
turn up the table s amplitude while keeping the forcing frequency xed
Initially the ball will remain stuck to the table until the table s maxi
mum acceleration is greater than the earth s gravitational acceleration
g The table s acceleration is given by
s A sin t
The maximum acceleration is thus A WhenA is greater than
g the ball b ecomes unstuck and will y free from the table until its
next impact The phase at which the ball b ecomes initially unstuck
o ccurs when
g
g A sin arcsin
unstuck unstuck
A
Even in a system in which the table s maximum acceleration is
much greater than g the ball can b ecome stuck An in nite number of
impacts can o ccur in a nite stopping time The sum of the times
between impacts converges in a nite time much less than the table s
period T The ball gets stuck again at the end of this sequence of
impacts and moves with the table until it reaches the phase
unstuck
This typ e of sticking solution is an eventually p erio dic orbit After its
rst time of getting stuck it will exactly rep eat this pattern of getting
stuck and then released forever
CHAPTER BOUNCING BALL
Figure Sticking solutions in the b ouncing ball system
However these sticking solutions are a bit exotic in several resp ects
Sticking solutions are not invertible that is an in nite numb er of initial
conditions can eventually arrive at the same identical sticking solution
It is imp ossible to run a sticking solution backward in time to nd the
exact initial condition from which the orbit came This is b ecause of
the geometric convergence of sticking solutions in nite time
Also there are an in nite numb er of di erent sticking solutions
Three such solutions are illustrated in Figure Toseehow some of
these solutions are formed let s turn the table amplitude up a little so
that the stopping time is lengthened Now it happ ens that the ball
do es not get stuck in the rst table p erio d T butkeeps b ouncing on
into the second or third p erio d However as it enters each new period
the b ounces get progressively lower so that the ball do es eventually get
stuck after several p erio ds Once stuck it again gets released when
the table s acceleration is greater than g and this new pattern rep eats
itself forever
QUALITATIVE DESCRIPTION OF MOTIONS
Figure Convergence to a p erio d one orbit
Figure Perio d two orbit of a b ouncing ball
Perio d One Orbits and Perio d Doubling
As we increase the table s amplitude we often see that the orbit jumps
from a sticking solution to a simple p erio dic motion Figure shows
the convergence of a tra jectory of the b ouncing ball system to a p erio d
one orbit The ball s motion converges toward a p erio dic orbit with a
p erio d exactly equal to that of the table hence the term periodone
orbit see Prob
What happ ens to the p erio d one solution as the forcing amplitude
of the table increases further We discover that the p erio d one orbit
bifurcates literally splits in two to the p erio d two orbit illustrated
in Figure Now the ball s motion is still p erio dic but it b ounces
high then low then high again requiring twice the table s p erio d to
complete a full cycle If we gradually increase the table s amplitude still
further we next discover a p erio d four orbit and then a p erio d eight
orbit and so on In this period doubling cascade we only see orbits of
period
n
P
CHAPTER BOUNCING BALL
Figure Chaotic orbit of a b ouncing ball
and not for instance p erio d three ve or six
n
The amplitude ranges for whicheach of these p erio d orbits is
observable however gets smaller and smaller Eventually it converges
to a critical table amplitude b eyond which the b ouncing ball system
exhibits the nonp erio dic b ehavior illustrated in Figure This last
typ e of motion found at the end of the p erio d doubling cascade never
settles down to a p erio dic orbit and is in fact our rst physical example
ofachaotic tra jectory known as a strange attractor This motion is
an attractor b ecause it is the asymptotic solution arising from many
di erent initial conditions di erent motions of the system are attracted
to this particular motion Atthispoint the term strange is used to
distinguish this motion from other motions such as p erio dic orbits or
equilibrium p oints A more precise de nition of the term strange is
given in section
At still higher table amplitudes many other typ es of strange and
p erio dic motions are p ossible a few of which are illustrated in Fig
ure The typ e of motion dep ends on the system parameters and
the sp eci c initial conditions It is common in a nonlinear system for
many solutions to co exist That is it is p ossible to see several di erent
p erio dic and chaotic motions for the same parameter values These
co existing orbits are only distinguished by their initial conditions
The p erio d doubling route to chaos wesawabove is common to
awidevariety of nonlinear systems and will b e discussed in depth in section
QUALITATIVE DESCRIPTION OF MOTIONS
Figure A zo o of p erio dic and chaotic motions seen in the b ouncing
ball system Generated by the Bouncing Ball program
Chaotic Motions
Figure shows the impact map of the strange attractor discovered
at the end of the p erio d doubling route to chaos This strange attractor
lo oks almost like a simple curve segment of an upside down parab ola
with gaps Parts of this curvelookchopp ed out or eaten away However
on magni cation this curve app ears not so simple after all Rather it
seems to resemble an intricate web of p oints spread out on a narrow
curved strip Since this chaotic solution is not p erio dic and hence
never exactly rep eats itself it must consist of an in nite collection of
discrete p oints in the impact velo city vs phase space
CHAPTER BOUNCING BALL
Figure Strange attractor in the b ouncing ball system arising at
the end of a p erio d doubling cascade Generated by the Bouncing Ball
program
This strange set is generated by an orbit of the b ouncing ball system
and it is chaotic in that orbits in this set exhibit sensitive dep endence
on initial conditions This sensitive dep endence on initial conditions is
easy to see in the b ouncing ball system when wesolve for the impact
phases and velo cities for the exact mo del with the numerical pro cedure
describ ed in App endix A First consider two slightly di erent tra jecto
ries that converge to the same p erio d one orbit As shown in Table
these orbits initially di er in phase by This phase di erence
increases a little over the next few impacts but by the eleventh impact
the orbits are indistinguishable from each other and by the eighteenth
impact they are indistinguishable from the p erio d one orbit Thus the
di erence b etween the two orbits decreases as the system evolves
An attracting p erio dic orbit has b oth long term and short term pre
dictability As the last example indicates we can predict from an ini
QUALITATIVE DESCRIPTION OF MOTIONS
Hit Phase Phase
Table Convergence of two di erent initial conditions to a p erio d one
orbit The digits in b old are where the orbits di er At the zeroth hit
the orbits di er in phase by Note that the di erence b etween
the orbits decreases so that after impacts b oth orbits are indistin
guishable from the p erio d one orbit The op erating parameters are
A cm frequency Hz and the initial ball velo city
is cm s The impact phase is presented as mo d
so that it is normalized to b e b etween zero and one
CHAPTER BOUNCING BALL
Hit Phase Phase
Table Divergence of initial conditions on a strange attractor il
lustrating sensitive dep endence on initial conditions The parameter
values are the same as in Table except for A cm The b old
digits show where the impact phases di er The orbits di er in phase
by at the zeroth hit but bythetwenty third impact they di er
at every digit
ATTRACTORS
tial condition of limited resolution where the ball will b e after a few
b ounces short term and after many b ounces long term
The situation is dramatically di erent for motion on a strange at
tractor Chaotic motions may still p ossess short term predictability
but they lack long term predictability InTable weshowtwodif
ferent tra jectories that again di er in phase by at the zeroth
impact However in the chaotic case the di erence increases at a re
markable rate with the evolution of the system That is given a small
di erence in the initial conditions the orbits diverge rapidly By the
twelfth impact the error is greater than bythetwentieth impact
and by the twenty fourth impact the orbits show no resemblance
Chaotic motion thus exhibits sensitive dep endence on initial conditions
Even if we increase our precision we still cannot predict the orbit s fu
ture p osition exactly
As a practical matter wehave no exact long term predictivepower
for chaotic motions It do es not really help to double the resolution of
our initial measurement as this will just p ostp one the problem The
b ouncing ball system is b oth deterministic and unpredictable Chaotic
motions of the b ouncing ball system are unpredictable in the practical
sense that initial measurements are always of limited accuracy and
any initial measurement error grows rapidly with the evolution of the
system
Strange attractors are common to a wide variety of nonlinear sys
tems We will develop a way to name and dissect these critters in
Chapters and
Attractors
An attracting set A in a trapping region D is de ned as a nonempty
closed set formed from some op en neighborhood
n
A f D
n
Wementioned b efore that for a dissipative b ouncing ball system the
trapping region is contracting so the op en neighborhood typically con
sists of a collection of smaller and smaller regions as it approaches the attracting set
CHAPTER BOUNCING BALL
Figure A p erio dic attractor and its transient
An attractor is an attempt to de ne the asymptotic solution of a
dynamical system It is that part of the solution that is left after the
transient part is thrown out Consider Figure whichshows the
approach of several phase space tra jectories of the b ouncing ball system
toward a p erio d one cycle The orbits app ear to consist of two parts
the transient the initial part of the orbit that is spiraling toward a
closed curve and the attractor the closed p erio dic orbit itself
In the previous section wesaw examples of several di erenttyp es
of attractors For small table amplitudes the ball comes to rest on
the table For these equilibrium solutions the attractor consists of a
single p oint in the phase space of the table s reference frame At higher
table amplitudes p erio dic orbits can exist in which case the attractor
is a closed curve in phase space In a dissipative system this closed
curve representing a p erio dic motion is also known as a limit cycle At
still higher table amplitudes a more complicated set called a strange
attractor can app ear The phase space plot of a strange attractor is a
complicated curve that never quite closes After a long time this curve
app ears to sketch out a surface Eachtyp e of attractor a p oint closed
curve or strange attractor something b etween a curve and a surface
represents a di erenttyp e of motion for the system equilibrium pe
rio dic or chaotic
Except for the equilibrium solutions each of the attractors just
BIFURCATION DIAGRAMS
describ ed in the phase space has its corresp onding representation in
the impact map In general the representation of the attractor in the
impact map is a geometric ob ject of one dimension less than in phase
space For instance a p erio dic orbit is a closed curve in phase space
and this same p erio d n orbit consists of a collection of n points in the
impact map The impact map for a chaotic orbit consists of an in nite
collection of p oints
For a nonlinear system many attractors can co exist This naturally
raises the question as to which orbits and collections of initial conditions
go to which attractors For a given attractor the domain of attraction
or basin of attraction is the collection of all those initial conditions
whose orbits approach and always remain near that attractor That
is it is the collection of all orbits that are captured by an attractor
Like the attractors themselves the basins of attraction can b e simple
or complex
Figure shows a diagram of basins of attraction in the b ouncing
ball system The phase space is dominated by the black regions which
indicate initial conditions that eventually b ecome sticking solutions
The white sinusoidal regions at the b ottom of Figure showun
physical initial conditions phases and velo cities that the ball cannot
obtain The gray regions represent initial conditions that approacha
p erio d one orbit See Plate for a color diagram of basins of attraction
in the b ouncing ball system
Bifurcation Diagrams
A bifurcation diagram provides a nice summary for the transition b e
tween di erenttyp es of motion that can o ccur as one parameter of the
system is varied A bifurcation diagram plots a system parameter on
the horizontal axis and a representation of an attractor on the vertical
axis For instance for the b ouncing ball system a bifurcation diagram
can show the table s forcing amplitude on the horizontal axis and the
asymptotic value of the ball s impact phase on the vertical axis as il
lustrated in Figure At a bifurcation p oint the attracting orbit
undergo es a qualitativechange For instance the attractor literally
splits in two in the bifurcation diagram when the attractor changes
CHAPTER BOUNCING BALL
Figure Basins of attraction in the b ouncing ball system Gener
ated by the Bouncing Ball program
BIFURCATION DIAGRAMS
Figure Bouncing ball bifurcation diagram Generated by the
Bouncing Ball program
from a p erio d one orbit to a p erio d two orbit
This b ouncing ball bifurcation diagram Fig shows the classic
p erio d doubling route to chaos For table amplitudes b etween cm
and cm a stable p erio d one orbit exists the ball impacts with
the table at a single phase For amplitudes b etween cm and
cm a p erio d two orbit exists The ball hits the table at two
distinct phases At higher table amplitudes the ball impacts at more
and more phases The ball hits at an in nity of distinct p oints phases
when the motion is chaotic
Bouncing Ball
References and Notes
For some recent examples of engineering applications in which the b ouncing ball
problem arises see M O Hongler P Cartier and P Flury Numerical study
of a mo del of vibro transp orter Phys Lett A M O
Hongler and J Figour Perio dic versus chaotic dynamics in vibratory feeders
Helv Phys Acta T O Dalrymple Numerical solutions to
vibroimpact via an initial value problem formulation J Sound Vib
The b ouncing ball problem has proved to b e a useful system for exp erimentally
exploring several new nonlinear e ects Examples of some of these exp eriments
include S Celaschi and R L Zimmerman Evolution of a two parameter chaotic
dynamics from universal attractors Phys Lett A
N B Tu llaro T M Mello Y M Choi and A M Albano Perio d dou
bling b oundaries of a b ouncing ball J Phys Paris K
Wiesenfeld and N B Tu llaro Suppression of p erio d doubling in the dynam
ics of the b ouncing ball Physica D P Pieranski Jumping
particle mo del Perio d doubling cascade in an exp erimental system J Phys
Paris P Pieranski Z Kowalik and M Franaszek Jump
ing particle mo del A study of the phase of a nonlinear dynamical system b elow
its transition to chaos J Phys Paris M Franaszek and
P Pieranski Jumping particle mo del Critical slowing down near the bifurca
tion p oints Can J Phys P Pieranski and R Bartolino
Jumping particle mo del Mo dulation mo des and resonant resp onse to a p eri
o dic p erturbation J Phys Paris M Franaszek and Z
J Kowalik Measurements of the dimension of the strange attractor for the
Fermi Ulam problem Phys Rev A P Pieranski and
J Ma lecki Noisy precursors and resonant prop erties of the p erio d doubling
mo des in a nonlinear dynamical system Phys Rev A
P Pieranski and J Ma lecki Noise sensitivehysteresis lo ops and p erio d dou
bling bifurcation p oints NuovoCimento D P Pieranski
Direct evidence for the suppression of p erio d doubling in the b ouncing ball
mo del Phys Rev A Z J Kowalik M Franaszek
and P Pieranski Self reanimating chaos in the b ouncing ball system Phys
Rev A
A description of some simple exp erimental b ouncing ball systems suitable for
undergraduate labs can b e found in A B Pippard The physics of vibration
Vol New York Cambridge University Press p N B Tu llaro
and A M Albano Chaotic dynamics of a b ouncing ball Am J Phys
R L Zimmerman and S Celaschi Comment on Chaotic
dynamics of a b ouncing ball Am J Phys T
M Mello and N B Tu llaro Strange attractors of a b ouncing ball Am J
Phys R Minnix and D Carp enter Piezo electric lm
Problems
reveals f versus t of ball b ounce The Physics Teacher March
The KYNAR piezo electric lm used for the impact detector is available from
Pennwalt Corp oration First Avenue P O Box C King of Prussia PA
The exp erimenter s kit containing an assortment of piezo electric
lms costs ab out
Some approximate mo dels of the b ouncing ball system are presented byG M
Zaslavsky The simplest case of a strange attractor Phys Lett A
P J Holmes The dynamics of rep eated impacts with a sinusoidally
vibrating table J Sound Vib M Franaszek E ect of
random noise on the deterministic chaos in a dissipative system Phys Lett
A R M Everson Chaotic dynamics of a b ouncing
ball Physica D A Mehta and J Luck Novel temp oral
behavior of a nonlinear dynamical system The completely inelastic b ouncing
ball Phys Rev Lett
A detailed comparison b etween the exact mo del and the high b ounce mo del is
given by C N Bapat S Sankar and N Popplewell Rep eated impacts on a
sinusoidally vibrating table reappraised J Sound Vib
A D Bernstein Listening to the co e cient of restitution Am J Phys
H M Isom aki Fractal prop erties of the b ouncing ball dynamics in Nonlinear
dynamics in engineering systems edited byW Schiehlen Springer Verlag
New York pp
Problems
Problems for section
For a p erio d one orbit in the exact mo del showthat
a v v
k
k
b v gT
k
c the impact phase is exactly given by
gT
cos
P
A
Bouncing Ball
Assuming only that v v
k
k
a Show that the solution in Problem can b e generalized to an nth
order symmetric p erio dic equispaced orbit satisfying v ng T for
k
n
b Show that the impact phase is exactly given by
ng T
cos
Pn
A
c Draw pictures of a few of these orbits Why are they called equispaced
d Find parameter values at which a p erio d one and p erio d two n
equispaced orbit can co exist
If T s and what is the smallest table amplitude for whicha
p erio d one orbit can exist use Prob c For these parameters estimate
the maximum height the ball b ounces and express your answer in units of the
average thicknessofahuman hair Describ e howyou arrived at this thickness
Describ e a numerical metho d for solving the exact b ouncing ball map How
do you determine when the ball gets stuck Howdoyou prop ose to nd the
zeros of the phase map equation
Derive equation from equation
Section
Calculate in equation for n when
n n
and
Con rm that the variables and given by equations are
dimensionless
Verify the derivation of equation the standard map
Calculate the inverse of the standard map eq
Write a computer program to iterate the mo del of the b ouncing ball system
given by equation the high b ounce approximation
Section
a Calculate the stopping time eq for a rst impact time
and a damping co e cient Also calculate and
b Calculate and the stopping time for an initial velo cityof m s
cm s and a damping co e cient of
Problems
For the high b ounce approximation eq show that when
a j j j j
j j
b A trapping region is given by a strip b ounded by v where v
max max
Note The reader may assume as the b o ok do es at the end of
section that the v cannot approach v asymptotically and that once
i max
inside the strip the orbit cannot leave
Calculate eq for A cm and T s What is the
unstuck
sp eed and acceleration of the table at this phase Is the table on its wayup
or down Are there table parameters for which the ball can b ecome unstuck
when the table is moving up Are there table parameters for which the ball
can b ecome unstuck when the table is moving down
Show that the average table velo citybetween impacts equals the average ball
velo citybetween impacts
These problems relate to the trapping region discussion in section
a Prove that v cannot approachthe v given by equation asymp
i max
totically Hint It is acceptable to increase v by some small
max
b Prove that for the trapping region D given by equation once the
orbit enters D it can never escap e D
c Use the trapping region D in the impact map to nd a trapping region in
phase space Hint Use the maximum outgoing velo city v to calculate a
max
minimum incoming velo city and a maximum height
d The trapping region found in the text is not unique in fact it is fairly
lo ose Try to obtain a smaller tighter trapping region
Derive equation
Section
How many p erio d one orbits can exist according to Problem c and how
many of these p erio d one orbits are attractors
Section
Write a computer program to generate a bifurcation diagram for the b ouncing ball system
Bouncing Ball
Chapter
Quadratic Map
Intro duction
A ball b ouncing on an oscillating table gives rise to complicated phe
nomena which app ear to defy our comprehension and analysis The
motions in the b ouncing ball system are truly complex However part
of the problem is that wedonot asyet havetheright language with
which to discuss nonlinear phenomena Wethus need to develop a
vo cabulary for nonlinear dynamics
A go o d rst step in developing any scienti c vo cabulary is the de
tailed analysis of some simple examples In this chapter we will b egin
by exploring the quadratic map In linear dynamics the corresp onding
example used for building a scienti c vo cabulary is the simple harmonic
oscillator see Figure As its name implies the harmonic oscillator
is a simple mo del which illustrates manykey notions useful in the study
of linear systems The image of a mass on a spring is usually not far
from one s mind even when dealing with the most abstract problems in
linear physics
The similarities among linear systems are easy to identify b ecause
of the extensive development of linear theory over the past century
Casual insp ection of nonlinear systems suggests little similarity Care
ful insp ection though reveals many common features Our original
intuition is misleading b ecause it is steep ed in linear theory Nonlin
ear systems p ossess as many similarities as di erences However the
CHAPTER QUADRATIC MAP
Figure Simple harmonic oscillator
vo cabulary of linear dynamics is inadequate to name these common
structures Thus our task is to discover the common elements of non
linear systems and to analyze their structure
A simple mo del of a nonlinear system is given by the di erence
equation known as a quadratic map
x x x
n n
n
x x
n n
For instance if wesetthevalue and initial condition x
in the quadratic map we nd that
x
x
x
etc
and in this case the value x app ears to b e approaching
n
Phenomena illustrated in the quadratic map arise in a wide variety
of nonlinear systems The quadratic map is also known as the logistic
INTRODUCTION
map and it was studied as early as byP F Verhulst as a mo del
for p opulation growth Verhulst was led to this di erence equation by
the following reasoning Supp ose in anygiven year indexed by the
subscript n that the normalized p opulation is x Then to nd the
n
p opulation in the next year x it seems reasonable to assume that
n
the numb er of new births will b e prop ortional to the current p opulation
x and the remaining inhabitable space x The pro duct of these
n n
two factors and gives the quadratic map where is some parameter
that dep ends on the fertility rate the initial living area the average
disease rate and so on
Given the quadratic map as our mo del for p opulation dynamics it
would now seem like an easy problem to predict the future p opulation
Will it grow decline or vary in a cyclic pattern As wewillsee the
answer to this question is easy to discover for some values of butnot
for others The dynamics are di cult to predict b ecause in addition
to exhibiting cyclic b ehavior it is also p ossible for the p opulation to
vary in a chaotic manner
In the context of physical systems the study of the quadratic map
was rst advo cated by E N Lorenz in At the time Lorenz was
lo oking at the convection of air in certain mo dels of weather prediction
Lorenz was led to the quadratic map by the following reasoning which
also applies to the b ouncing ball system as well as to Lorenz s original
mo del or for that matter to any highly dissipative system Consider
a time series that comes from the measurementofavariable in some
physical system
fx x x x x x x g
i n n
For instance in the b ouncing ball system this time series could consist
of the sequence of impact phases so that x x x
and so on We require that this time series arise from motion on an
attractor To meet this requirement we throwoutany measurements
that are part of the initial transient motion In addition we assume
no foreknowledge of how to mo del the pro cess giving rise to this time
series Given our ignorance it then seems natural to try to predict the
n st element of the time series from the previous nth value Formally
CHAPTER QUADRATIC MAP
we are seeking a function f such that
x f x
n n
In the b ouncing ball example this idea suggests that the next impact
phase would b e a function of the previous impact phase that is
n
f
n
If such a simple relation exists then it should b e easy to see by
plotting y x on the vertical axis and x on the horizontal axis
n n n
Formally we are taking our original time series equation and cre
ating an embedded time series consisting of the ordered pairs x y
x x
n n
f x x x x x x x x x x g
n n n n
Theideaofemb edding a time series will b e central to the exp erimental
study of nonlinear systems discussed in section In Figure we
showanemb edded time series of the impact phases for chaotic motions
in the b ouncing ball system The p oints for this emb edded time series
app ear to lie close to a region that resembles an upside down parab ola
The exact details of the curve dep end of course on the sp eci c param
eter values but as a rst approximation the quadratic map provides
a reasonable t to this curve see Figure Note that the curve s
maximum amplitude lo cated at the p oint x rises as the param
eter increases We think of the parameter in the quadratic map as
representing some parameter in our pro cess could b e analogous to
the table s forcing amplitude in the b ouncing ball system Such single
hump ed maps often arise when studying highly dissipative nonlinear
systems Of course more complicated many hump ed maps can and do
o ccur however the single hump ed map is the simplest and is therefore
a go o d place to start in developing our new vo cabulary
Iteration and Di erentiation
In the previous section weintro duced the equation
f x x x
ITERATION AND DIFFERENTIATION
Figure Emb edded time series of chaotic motion in the b ouncing
ball system Generated by the Bouncing Ball program
CHAPTER QUADRATIC MAP
Figure The quadratic function Generated by the Quadratic Map program
ITERATION AND DIFFERENTIATION
known as the quadratic map We write f x when wewanttomake
the dep endence of f on the parameter explicit
In this section we review two mathematical to ols we will need for
the rest of the chapter iteration and di erentiation We think of a
map f x x as generating a sequence of p oints With the seed
n n
n
x de ne x f x and consider the sequence x x x x as
n
an orbit of the map That is the orbit is the sequence of p oints
x x f x x f x x f x
where the nth iterate of x is found by functional comp osition n times
f f f
f f f f
n
z
n
f f f f f
When determining the stability of an orbit wewillneedtocalcu
late the derivative of these comp osite functions see section The
derivative of a comp osite function evaluated at a p oint x x is written
as
d
n n
f x f x j
x x
dx
The left hand side of equation is a shorthand form for the right
hand side that tells us to do the following when calculating the deriva
n
tive First construct the nth comp osite function of f call it f Sec
n
ond compute the derivativeof f And third as the bar notation
j tells us evaluate this derivativeat x x For instance if
x x
f x x n and x then
d
f x j f
x
dx
d d d
f f x f x x
dx dx dx
x j
x
Notice that we suppressed the bar notation during the intermediate
steps This is common practice when the meaning is clear from context
CHAPTER QUADRATIC MAP
You may sometimes see the even shorter notation for evaluating the
derivativeata point x as
d
n n
f x j f x
x x
dx
which is su ciently terse to b e legitimately confusing
An examination of the dynamics of the quadratic map provides an
excellentintro duction to the rich b ehavior that can exist in nonlinear
systems To nd the itinerary of an individual orbit all we need is a
pocket calculator or a computer program something like the following
C program
quadratic c calculate an orbit for the quadratic map
input l x
output x
x
x
etc
include stdio h
main
f
int n
float lambda x zero x n
zeronn scanf f f lambda x zero printf Enter lambda x
x n xzero
for n n n f
x n lambda x n xn the quadratic map
n printf d fnn n x
g
g
A nice brief intro duction to the C programming language su cient for most of
the programs in this b o ok is Chapter A tutorial intro duction of B W Kernighan
andD M Ritchie TheCProgramming Language Prentice Hall Englewo o d Cli s NJ pp
GRAPHICAL METHOD
Figure Graphical metho d for iterating the quadratic map Gen
erated by the Quadratic Map program
Graphical Metho d
In addition to doing a calculation there is a graphical pro cedure for
nding the itinerary of an orbit This graphical metho d is illustrated
in Figure for the same parameter value and initial condition used
in the previous example and is based on the following observation To
nd x from x we note that x f x graphically to get f x
n n n n n
we start at x on the horizontal axis and movevertically until we hit
n
the graph y f x Now this currentvalue of y must b e transferred
from the vertical axis back to the horizontal axis so that it can b e used
as the next seed for the quadratic map The simplest way to transfer
the y axis to the x axis is by folding the x y plane through the diagonal
line y x since p oints on the vertical axis are identical to p oints on
the horizontal axis on this line This insight suggests the following
graphical recip e for nding the orbit for some initial condition x
CHAPTER QUADRATIC MAP
Figure Graphical iteration of the quadratic map for a chaotic orbit
Generated by the Quadratic Map program
Start at x on the horizontal axis
Movevertically up until you hit the graph f x
Move horizontal ly until you hit the diagonal line y x
Move vertical ly up or down until you hit the graph f x
Rep eat steps and to generate new p oints
For the example in Figure it is clear that the orbit is converging
to the p oint This same graphical technique is also illustrated in
Figure for the more complicated orbit that arises when
We can also ask again ab out the fate of a whole collection of initial
conditions instead of just a single orbit In particular we can consider
the transformation of all initial conditions on the unit interval
I fxjx g
GRAPHICAL METHOD
Figure Stretching and folding in the quadratic map
sub ject to the quadratic map equation As shown in Figure
the quadratic map for canbeviewed as transforming the unit
interval in two steps The rst step is a stretch whichtakes the interval
to twice its length f The second step is a fold
str etch
whichtakes the lower half of the interval to the whole unit interval
and the upp er half of the interval also to the whole unit interval with
its direction reversed f and These
f old
two op erations of stretching and folding are the key geometric construc
tions leading to the complex b ehavior found in nonlinear systems The
stretching op eration tends to quickly separate nearby p oints while the
folding op eration ensures that all p oints will remain b ounded in some
region of phase space
Another way to visualize this stretching and folding pro cess is pre
sented in Figure Imagine taking a rubb er sheet and dividing it into
two sections by slicing it down the middle The sheet separates into
two branches at the gap the upp er part of the sheet where the slice
b egins the left branch is at while the right branch has a half twist
in it These two branches are rejoined or glued together again at the
branch line seen at the b ottom of the diagram Notice that the left
branch passes b ehind the right branch Next imagine that there is a
simple rule indicated by the wide arrows in the diagram that smo othly
carries p oints at the top of the sheet to p oints at the b ottom In partic
ular the unit interval at the top of the sheet gets stretched and folded
so that it ends up as the b ent line segment indicated at the b ottom
CHAPTER QUADRATIC MAP
Figure Rubb er sheet mo del of stretching and folding in the
quadratic map
of the sheet Atthispoint Figure is just a visual aid illustrating
how stretching and folding can o ccur in a dynamical system However
observe that the resulting folded line segment resembles a horsesho e
These horsesho es were rst identi ed and analyzed as recurring ele
ments in nonlinear systems by the mathematician Steve Smale We
will saymuch more ab out these horsesho es and make more extensive
use of such diagrams in section and Chapter when we try to
unravel the top ological organization of strange sets
Fixed Points
A simple linear map f R R of the real line R to itself is given
by f x mx Unlike the quadratic map this linear map can have
stretching but no folding The graphical analysis shown in Figure
quickly convinces us that for x only three p ossible asymptotic
states exist namely
lim x if m
n n
lim x if m and
n n
x x for m
n n
FIXED POINTS
Figure Graphical iteration of the linear map
The p erio d one p oints of a map p oints that map to themselves after
one iteration are also called xedpoints If m then the origin
is an attracting xedpoint or sink since nearbypoints tend to see
Figure a If m then the origin is still a xed p oint However
b ecause p oints near the origin always tend away from it the origin is
called a repel ling xedpoint or source see Figure b Lastly if
m then all initial conditions lead immediately to a p erio d one
orbit de ned by y x All the p erio dic orbits that lie on this line have
neutral stability
The story for the more complicated function f x x is not much
di erent For this parab olic map a simple graphical analysis shows that
as n
n
f x if jxj
n
f x if jxj
n
f for all n
n
f if n
In this case all initial conditions tend to either or except for the
p oint x which is a rep elling xed p oint since all nearby orbits
moveaway from The sp ecial initial condition x is said to b e
eventual ly xed b ecause although it is not a xed p oint itself it go es
exactly to a xed p oint in a nite numb er of iterations The sticking
solutions of the b ouncing ball system are examples of orbits that could
b e called eventually p erio dic since they arrive at a p erio dic orbit in a nite time
CHAPTER QUADRATIC MAP
Figure The lo cal stability of a xed p oint is determined bythe
slop e of f at x
Graphical analysis also allows us to see why certain xed p oints are
lo cally attracting and others rep elling As Figure illustrates the
lo cal stabilityofa xedpoint is determined by the slop e of the curve
passing through the xed p oint If the absolute value of the slop e is less
than one or equivalently if the absolute value of the derivativeatthe
xed p oint is less than one then the xed p oint is lo cally attracting
Alternatively if the absolute value of the derivative at the xed p oint
is greater then one then the xed p oint is rep elling
An orbit of a map is p erio dic if it rep eats itself after a nite number
of iterations For instance a p ointonaperiodtwo orbit has the prop
erty that f x x and a p erio d three p oint satis es f x x
that is it rep eats itself after three iterations In general a p erio d n
point rep eats itself after n iterations and is a solution to the equation
n
f x x
PERIODIC ORBITS
In other words aperiodnpoint is a xedpoint of the nth composite
function of f Accordingly the stabilityofthis xedpoint and of the
n
corresp onding p erio d n orbit is determined by the derivativeof f x
Our discussion ab out the xed p oints of a map is summarized in
the following two de nitions concerning xed p oints p erio dic p oints
and their stability A more rigorous account of p erio dic orbits and
their stability in presented in section
De nition Let f R R The p oint x is a xedpoint for f
if f x x The p oint x is a periodic point of p erio d n for f if
n i
f x x but f x x for i n The p oint x is eventual ly
m m n
periodic if f x f x but x is not itself p erio dic
n
De nition A p erio dic p oint x of period n is attracting if j f x j
The prime denotes di erentiation with resp ect to x The p eri
n
odic point x is repel ling if j f x j The p oint x is neutral if
n
j f x j
Wehavejustshown that the dynamics of the linear map and the
parab olic map are easy to understand By combining these two maps
we arrive at the quadratic map which exhibits complex dynamics The
quadratic map is then in a way the simplest map exhibiting nontrivial
nonlinear b ehavior
Perio dic Orbits
From our de nition of a p erio d n p oint namely
n
f x x
we see that nding a p erio d n orbit for the quadratic map requires
n
nding the zeros for a p olynomial of order For instance the p erio d
one orbits are given by the ro ots of
f x x x x
which is a p olynomial of order The p erio d two orbits are found by
evaluating
f x f f x
CHAPTER QUADRATIC MAP
f x x
x x x x x
which is a p olynomial of order Similarly the p erio d three orbits are
given by solving a p olynomial of order and so on Unfortunately
except for small n solving such high order p olynomials is b eyond the
means of b oth mortals and machines
Furthermore our de nition for the stabilityofanorbitsays that
once we nd a p oint of a p erio d n orbit call it x wenextneedto
evaluate the derivative of our p olynomial at that p oint For instance
the stability of a p erio d one orbit is determined byevaluating
d
x x x j f x
x x
dx
Similarly the stability of a p erio d two orbit is determined from the
equation
d
f x x x x x j
x x
dx
x x x
Again these stability p olynomials quickly b ecome to o cumb ersome to
analyze as n increases
Any p erio dic orbit of p erio d n will have n points in its orbit We will
generally lab el this collection of p oints by the subscript i
n so that
g x x x x x x fx
i
n n
where i lab els an individual p oint of the orbit The b oldface notation
indicates that x is an n tuple of real numb ers Another complication
will arise in some cases it is useful to write our indexing subscript in
some base other than ten For instance it is useful to workinbasetwo
when studying one hump ed maps In general it is convenienttowork
in base n where n is the numb er of critical p oints of the map It
will b e advantageous to lab el the orbits in the quadratic map according
to some binary scheme
PERIODIC ORBITS
Lastly the question arises whichelement of the p erio dic orbit do
we use in evaluating the stability of an orbit In Problem we
show that al l periodic points in a periodic orbit give the same value
n
for the stability function f So we can use any p ointinthe
p erio dic sequence This fact is go o d to keep in mind when evaluating
the stability of an orbit
Graphical Metho d
Although the algebra is hop eless the geometric interpretation for the
lo cation of p erio dic orbits is straightforward As we see in Figure
a the lo cation of the p erio d one orbits is given bytheintersection
of the graphs y f x and y x The latter equation is simply a
straight line passing through the origin with slop e In the case of
the quadratic map f is an inverted parab ola also passing through the
origin These two graphs can intersect at two distinct p oints giving
rise to two distinct p erio d one orbits One of these orbits is always at
the origin and the other s exact lo cation dep ends on the height of the
quadratic map that is the sp eci c value of in the quadratic map
To nd the lo cation of the p erio d two orbit weneedtoplot y x
and f x The graph shown in Figure b shows three p oints of in
tersection in addition to the origin The middle p oint the op en circle
is the p erio d one orbit found ab ove The two remaining intersection
p oints are the twopoints b elonging to a single p erio d two orbit A
dashed line indicates where these p erio d two p oints sit on the original
quadratic map the two dark circles and the simple graphical con
struction of section should convince the reader that this is in fact
a p erio d two orbit
The story for higher order orbits is the same see Fig a and
b The graph of the third iterate y f x shows eight p oints
of intersection with the straight line Not all eightintersection p oints
are elements of a p erio d three orbit Twoofthesepoints are just the
pair of p erio d one orbits The remaining six p oints consist of a pair of
p erio d three orbits The graph for the p erio d four orbits shows sixteen
p oints of intersection Again not all the intersection p oints are part of
a p erio d four orbit Twointersection p oints are from the pair of p erio d
one orbits and two are from the p erio d two orbit That leaves twelve
CHAPTER QUADRATIC MAP
Figure First and second iterates of the quadratic map
PERIODIC ORBITS
Figure Third and fourth iterates of the quadratic map
CHAPTER QUADRATIC MAP
remaining p oints of intersection eachofwhich is part of some p erio d
four orbit Since there are twelve remaining p oints there must b e three
p oints p oints p er orbit distinct p erio d four orbits
n
The number of intersection p oints of f dep ends on If
and n there are only twointersection p oints the two distinct p e
rio d one orbits In dramatic contrast if then it is easy to show
n
that there will b e intersection p oints and counting arguments like
those just illustrated allow us to determine howmany of these intersec
tion p oints are new p erio dic p oints of p erio d n One fundamental
question is how can a system as simple as the quadratic map change
from having only twotohaving an in nite numb er of p erio dic orbits
Likemany asp ects of the quadratic map the answers are surprising
Before wetackle this problem let s resume our analysis of the p erio d
one and p erio d two orbits
Perio d One Orbits
Solving equation for x we ndtwo p erio d one solutions
x
and
x
The rst p erio d one orbit lab eled x always remains at the origin
while the lo cation of the second p erio d one orbit x dep ends on
From equation the stability of each of these orbits is determined
from
f x
The subscript n to x is lab eling two distinct periodic orbits This is p otentially
n
confusing notation since we previously reserved this subscript to lab el di erent
p oints in the same p erio dic orbit In practice this notation will not b e ambiguous
since this lab el will b e a binary index the length of which determines the p erio d
of the orbit Di erent cyclic p ermutations of this binary index will corresp ond to
di erentpoints on the same orbit A noncyclic p ermutation must then b e a p oint
on a distinct p erio d n orbit The rules for this binary lab eling scheme are sp elled out in section
PERIODIC ORBITS
and
f x
Clearly if then jf x j andjf x j so the p erio d
one orbit x is stable and x is unstable At these two orbits
collide and exchange stability so that for x is unstable and
x is stable For b oth orbits are unstable
Perio d Two Orbit
The lo cation of the p erio d two orbit is found from equation
p
x
and
p
x
These twopoints b elong to the p erio d two orbit We lab el the left p oint
x and the rightpoint x Note that the lo cation of the p erio d two
orbit pro duces complex numb ers for This indicates that the
period two orbit exists only for whichisobvious geometrically
since y f x b egins a new intersection with the straight line y x
at
The stability of this p erio d two orbit is determined by rewriting
equation as
f x x x
where we used equations and for x and x A plot of the
stability for the p erio d two orbit is presented in Figure A close
examination of this gure shows that for the absolute
value of the stability function is less than one that is the p erio d two
orbit is stable For the p erio d two orbit is unstable
The range in for which the p erio d two orbit is stable can actually
b e obtained analytically The p erio d two orbit is stable as long as
f x
The p erio d two orbit rst b ecomes stable when f x which
o ccurs at and it loses stabilityat f x whichthe
p
reader can verify takes place at see Prob
CHAPTER QUADRATIC MAP
Figure Stability of p erio d two orbit
Stability Diagram
The lo cation and stability of the two p erio d one orbits and the single
period two orbit are summarized in the orbit stability diagram shown
in Figure The vertical axis shows the lo cation of the p erio dic
as a function of the parameter Stable orbits are denoted by orbit x
n
solid lines unstable orbits by dashed lines Two bifurcation p oints
are evident in the diagram The rst o ccurs when the two p erio d one
orbits collide and exchange stabilityat The second o ccurs with
the birth of a stable p erio d two orbit from a stable p erio d one orbit at
Bifurcation Diagram
To explore the dynamics of the quadratic map further we can cho ose
an initial condition x and a parameter value and then iterate the
map using the program in section to see where the orbit go es We
would notice a few general results if weplay this game long enough
First if and x then the graphical analysis of section
shows us that all p oints not in the unit interval will run o to
BIFURCATION DIAGRAM
Figure Orbit stability diagram
in nity Further if then the same typ e of graphical analysis
shows that the dynamics of the quadratic map are simple there is only
one attracting xed p oint and one rep elling xed p oint These xed
p oints are the p erio d one orbits calculated in the previous section
Second the initial condition wepick is usually not imp ortantin
determining the attractor although the value of is very imp ortant
We seem to end up with the same attractor no matter what x
we pick The quadratic map usually has one and only one attractor
whereas most nonlinear systems can have more than one attractor
The b ouncing ball system for example can havetwo or more co existing
attractors
Third as wewillshow in section almost all initial conditions
run o to in nity for all There are no attractors in this case
Therefore when studying the quadratic map it will usually su ce
Some initial conditions do not converge to the attractor For instance any x
b elonging to an unstable p erio dic orbit will not converge to the attractor Unstable
orbits are by de nition not attractors so that almost any orbit near an unstable
p erio dic orbit will diverge from it and head toward some attractor
CHAPTER QUADRATIC MAP
n
to pick a single initial condition from the unit interval If f x ever
leaves the unit interval then it will run o to in nityandnever return
provided Further when studying attractors we can limit our
attention to values of If then the only attractor is
a stable xed p oint at zero and if there are no attractors
For all these reasons a bifurcation diagram is a particularly p owerful
metho d for studying the attractors in the quadratic map Recall that a
bifurcation diagram is a plot of an asymptotic solution on the vertical
axis and a control parameter on the horizontal axis To construct a
bifurcation diagram for the quadratic map only requires some simple
mo di cations of our previous program for iterating the quadratic map
As seen b elow the new algorithm consists of the following steps
Set and x almost any x will do
Iterate the quadratic map times to remove the transient
solution and then print and x for the next p oints
n
which are presumably part of the attractor
Increment by a small amount and set x to the last value
of x
n
Rep eat steps and until
A C program implementing this algorithm is as follows
bifquad c calculate bifurcation diagram for the quadratic map
input none
output l x
l x
etc
l x
l x
etc
include stdio h
main
f
int n
Technically the phase space of the quadratic map R can b e compacti ed
thereby making the p oint at in nitya valid attractor
BIFURCATION DIAGRAM
Figure Bifurcation diagram for the quadratic map
n float lambda x
x n
for lambda lambda lambda f
for n n n f
x n lambda x n xn
if n
n printf f fnn lambda x
g
g
g
When plotted in Figure for the output of our
simple program pro duces a bifurcation diagram of stunning complex
ity Ab ove the diagram we provide comments on the typ e of attractor
observed and on the horizontal axis signi cant parameter values are
indicated This bifurcation diagram shows many qualitative similari
ties to bifurcation diagrams from the b ouncing ball system compare
to Figure Both exhibit the p erio d doubling route to chaos For
the quadratic map an in nite numb er of p erio d doublings o ccur for
CHAPTER QUADRATIC MAP
Both also show p erio dic windows white bands within
the chaotic regions For the quadratic map a p erio d three window
begins at and a p erio d vewindow b egins at Lo ok
ing closely at the p erio dic windows we see that eachbranch of these
p erio dic windows also undergo es a p erio d doubling cascade
Bifurcation diagrams showing only attracting solutions can b e some
what misleading Much of the structure in the bifurcation diagram can
only b e understo o d bykeeping track of b oth the stable attracting solu
tions and the unstable rep elling solutions as we did in constructing the
orbit stability diagram Figure Just as there are stable p erio dic
orbits and chaotic attractors there are also unstable p erio dic orbits
and chaotic rep ellers In Figure for instance we indicate the ex
istence of an unstable p erio d one orbit by the dashed line b eginning
at the rst p erio d doubling bifurcation As we show in section
this unstable p erio d one orbit is simply the continuation of the stable
p erio d one orbit that exists b efore this p erio d doubling bifurcation see
Figures and
Lo cal Bifurcation Theory
Poincar e used the term bifurcation to describ e the splitting of asymp
totic states of a dynamical system Figure shows a bifurcation
diagram for the quadratic map see Plate for a color version of a
quadratic map bifurcation diagram As we examine Figure we
see that several di erenttyp es of changes can o ccur Wewould like
to analyze and classify these bifurcations At a bifurcation value the
qualitative nature of the solution changes It can change to or from an
equilibrium p erio dic or chaotic state It can change from one typ e of
p erio dic state to another or from one typ e of chaotic state to another
For instance in the b ouncing ball system we are initially in an
equilibrium state with the ball moving in unison with the table As
we turn up the table amplitude we rst nd sticking solutions As
we increase the amplitude further we nd a critical parameter value
at which the ball switches from the sticking b ehavior to b ouncing in a
p erio d one orbit Sucha change from an equilibrium state to a p erio dic
state is an example of a saddle no de bifurcation As we further turn
LOCAL BIFURCATION THEORY
Figure Saddle no de bifurcation diagram
up the table amplitude we nd that there is a second critical table
amplitude at which the ball switches from a p erio d one to a p erio d
two orbit The analogous p erio d doubling bifurcation in the quadratic
map o ccurs at Both the saddle no de and the p erio d doubling
bifurcations are examples of lo cal bifurcations At their birth or death
all the orbits participating are lo calized in phase space that is they all
start out close together Global bifurcations can also o ccur although
typically these are more di cult to analyze since they can give birth to
an in nite numb er of p erio dic orbits In this section we analyze three
simple typ es of lo cal bifurcations that commonly o ccur in nonlinear
systems These are the saddle no de p erio d doubling and transcritical
bifurcations
Saddle no de
In a sadd le node bifurcation a pair of p erio dic orbits are created out
of nothing One of the p erio dic orbits is always unstable the saddle
while the other p erio dic orbit is always stable the no de The basic
bifurcation diagram for a saddle no de bifurcation lo oks like that shown
in Figure The saddle no de bifurcation is fundamental to the study
of nonlinear systems since it is one of the most basic pro cesses bywhich p erio dic orbits are created
CHAPTER QUADRATIC MAP
Figure Tangency mechanism for a saddle no de bifurcation
A saddle no de bifurcation is also referred to as a tangent bifurcation
b ecause of the mechanism bywhich the orbits are b orn Consider the
nth comp osite of some mapping function f whichisneartoatangency
with the line y x Let b e the value at which a saddle no de
sn
n
is tangent to the line bifurcation o ccurs Notice in Figure that f
y x at For no p erio d n orbits exist in this neighborhood
sn sn
but for two orbits are b orn The lo cal stabilityofapointof
sn
n n
a map is determined by f Since f x is tangentto y x at a
bifurcation it follows that at
sn
n
f x
sn
Tangent bifurcations ab ound in the quadratic map For instance a
pair of p erio d three orbits are created by a tangent bifurcation in the
p
As illustrated in Figure quadratic map when
for there are eightpoints of intersection Twooftheinter
section p oints b elong to the p erio d one orbits while the remaining six
make up a pair of p erio d three orbits Near to tangency the absolute
value of the slop e at three of these p oints is greater than one this is the
unstable p erio d three orbit The remaining three p oints form the sta
ble p erio dic orbit The birth of this stable p erio d three orbit is clearly
visible as the p erio d three windowinournumerically constructed bi
furcation diagram of the quadratic map Figure In fact all the
See reference for more details
LOCAL BIFURCATION THEORY
Figure A pair of p erio d three orbits created by a tangentbi
furcation in the quadratic map shown with the unstable p erio d one
orbits
o dd p erio d orbits of the quadratic map are created by some sort of
tangent bifurcation
Perio d Doubling
Period doubling bifurcations are evidentwhenwe consider an even num
b er of comp ositions of the quadratic map In Figure weshow the
second iteration of the quadratic map near a tangency Belowthepe
rio d doubling bifurcation a single stable p erio d one orbit exists As
is increased the p erio d one orbit b ecomes unstable and a stable p erio d
two orbit is b orn This information is summarized in the bifurcation
diagram presented in Figure Let b e the parameter value at
pd
which the p erio d doubling bifurcation o ccurs At this parameter value
the p erio d one and the nascent p erio d two orbit coincide As illustrated
in Figure f x is tangenttoy x so that f x
pd pd
n
However f x that is at a p erio d doubling bifurcation the
pd
function determining the lo cal stability of the p erio dic orbit is always
Figure shows f just after p erio d doubling In general for a
CHAPTER QUADRATIC MAP
Figure Second iterate of the quadratic map near a tangency
Figure Perio d doubling ip bifurcation diagram
LOCAL BIFURCATION THEORY
Figure Tangency mechanism near a p erio d doubling ip bifur
cation
n
period n to p erio d n bifurcation f x and
pd
n
f x
pd
A p erio d doubling bifurcation is also known as a ip bifurcation In
the p erio d one to p erio d two bifurcation the p erio d two orbit ips from
side to side ab out its p erio d one parent orbit This is b ecause f x
pd
see Prob The rst ip bifurcation in the quadratic map
o ccurs at andwas analyzed in sections where we
considered the lo cation and stability of the p erio d one and p erio d two
orbits in the quadratic map
Transcritical
The last bifurcation we illustrate with the quadratic map is a trans
critical bifurcation inwhich an unstable and stable p erio dic orbit col
lide and exchange stability A transcritical bifurcation o ccurs in the
quadratic map when As in a saddle no de bifurcation
tc
f at a transcritical bifurcation However a transcritical bi
tc
furcation also has an additional constraint not found in a saddle no de
bifurcation namely
x f
tc
CHAPTER QUADRATIC MAP
For the quadratic map this xed p oint is just the p erio d one orbit at
the origin x found from equation
A summary of these three typ es of bifurcations is presented in Figure
Other typ es of lo cal bifurcations are p ossible a more complete
theory for b oth maps and ows is given in reference
Perio d Doubling Ad In nitum
A view of the bifurcation diagram for the quadratic map for between
and is presented in Figure This diagram reveals not
one but rather an in nite numb er of p erio d doubling bifurcations As
is increased a p erio d two orbit b ecomes a p erio d four orbit and
this in turn b ecomes a p erio d eight orbit and so on This sequence of
p erio d doubling bifurcations is known as a period doubling cascade This
pro cess app ears to converge at a nite value of around b eyond
which a nonp erio dic motion app ears to exist This p erio d doubling
cascade often o ccurs in nonlinear systems For instance a similar p erio d
doubling cascade o ccurs in the b ouncing ball system Figure The
p erio d doubling route is one common way but certainly not the only
way by which a nonlinear system can progress from a simple b ehavior
one or a few p erio dic orbits to a complex b ehavior chaotic motion
and the existence of an in nity of unstable p erio dic orbits
In Feigenbaum b egan to wonder ab out this p erio d doubling
cascade He started playing some numerical games with the quadratic
map using his HP hand held calculator His wondering so on led to a
remarkable discovery At the time Feigenbaum knew that this p erio d
doubling cascade o ccurred in one dimensional maps of the unit inter
val He also had some evidence that it o ccurred in simple systems of
nonlinear di erential equations that mo del for instance the motion of
a forced p endulum In addition to lo oking at the qualitative similarities
between these systems he b egan to ask if there mightbesome quan
titative similarity that is some numb ers that might b e the same in
all these di erent systems exhibiting p erio d doubling If these numbers
could b e found they would b e universal in the sense that they would
not dep end on the sp eci c details of the system
Feigenbaum was inspired in his search in part bya very success
PERIOD DOUBLING AD INFINITUM Figure Summary of bifurcations
CHAPTER QUADRATIC MAP
Figure Bifurcation diagram showing p erio d doubling in the
quadratic map
ful theory of universal numb ers for second order phase transitions in
physics A phase transition takes place in a system when a change of
state o ccurs During the s it was discovered that there were quan
titative measurements characterizing phase transitions that did not de
p end on the details of the substance used Moreover these universal
numb ers in the theory of phase transitions were successfully measured
in countless exp eriments throughout the world Feigenbaum wondered
if there might b e some similar universality theory for dissipative non
linear systems
Feigenbaum intro duced the renormalization group approach of critical phenom
ena to the study of nonlinear dynamical systems Additional early contributions
to these ideas came from Cvitanovi c and also Collet Coullet Eckmann Lanford
and Tresser The geometric convergence of the quadratic map was noted as early
as by Myrb erg see C Mira Chaotic dynamics World Scienti c New Jersey
and also by Grossmann and Thomae in
PERIOD DOUBLING AD INFINITUM
Table Perio d doubling bifurcation values for the quadratic map
By de nition suchuniversal numb ers are dimensionless the sp eci c
mechanical details of the system must b e scaled out of the problem
Feigenbaum b egan his search for universal numbers by examining the
p erio d doubling cascade in the quadratic map He recorded with the
help of his calculator the values of at which the rst few p erio d dou
bling bifurcations o ccur Wehave listed the rst eightvalues orbits up
to p erio d inTable While staring at this sequence of bifurcation
p oints Feigenbaum was immediately struckby the rapid convergence
of this series Indeed he recognized that the convergence app ears to
followthatofa geometric series similar to the one wesaw in equation
when we studied the sticking solutions of the b ouncing ball
Let be the value of the nth p erio d doubling bifurcation and
n
de ne as lim Based on his inspiration Feigenbaum guessed
n n
that this sequence ob eys a geometric convergence that is
n
c n
n
where c is a constant and is a constant greater than one Using
equation and a little algebra it follows that if we de ne by
n n
lim
n
n n
then is a dimensionless number characterizing the rate of convergence
of the p erio d doubling cascade
The three constants in this discussion have b een calculated as
and c
For a review of geometric series see anyintro ductory calculus text suchasC
Edwards and D Penny Calculus and analytic geometry Prentice Hall Englewood Cli s NJ p
CHAPTER QUADRATIC MAP
The constant is now called Feigenbaum s delta b ecause Feigenbaum
wentontoshowthatthisnumber is universal in that it arises in a
wide class of dissipative nonlinear systems that are close to the single
hump ed map This numb er has b een measured in exp eriments with
chicken hearts electronic circuits lasers chemical reactions and liquids
in their approach to a turbulent state as well as the b ouncing ball
system
To exp erimentally estimate Feigenbaum s delta all one needs to do
is measure the parameter values of the rst few p erio d doublings and
then substitute these numb ers into equation The geometric
convergence of is a mixed blessing for the exp erimentalist In practice
it means that converges very rapidly to so that only the rst
n
few s are needed to get a go o d estimate of Feigenbaum s delta It
n
also means that only the rst few s can b e exp erimentally measured
n
with any accuracy since the higher s bunchuptooquickly to
n
Tocontinue with more technical details of this story see Rasband s
account of renormalization theory for the quadratic map
Feigenbaum s result is remarkable in two resp ects Mathematically
he discovered a simple universal prop erty o ccurring in a wide class of
dynamical systems Feigenbaum s discovery is so simple and fundamen
tal that it could have b een made in or in for that matter
Still he had some help from his calculator It to ok a lot of numerical
work to develop the intuition that led Feigenbaum to his discovery and
it seems unlikely that the computational work needed would haveoc
curred without help from some sort of computational device suchasa
calculator or computer Physically Feigenbaum s result is remarkable
b ecause it p oints the waytoward a theory of nonlinear systems in which
complicated di erential equations whicheven the fastest computers
cannot solve are replaced by simple mo dels such as the quadratic
map which capture the essence of a nonlinear problem including its
solution The latter part of this story is still ongoing and there are
surely other gems to b e discovered with some inspiration p erspiration
and mayb e even a workstation
SARKOVSKI I S THEOREM
Sarkovskii s Theorem
In the previous section wesaw that for an in nite number
n
of p erio dic orbits with p erio d are b orn in the quadratic map In a
p erio d doubling cascade weknow the sequence in which these p erio dic
orbits are b orn A p erio d one orbit is b orn rst followed by a p erio d
two orbit a p erio d four orbit a p erio d eight orbit and so on For
higher values of additional p erio dic orbits come into existence For
p
as we instance a p erio d three orbit is b orn when
showed in section In this section we will explicitly show that all
p ossible p erio dic orbits exist for One of the goals of bifurcation
theory is to understand the di erentmechanisms for the birth and death
of these p erio dic orbits Pinning down all the details of an individual
problem is usually very di cult often imp ossible However there is
one qualitative result due to Sarkovskii of great b eauty that applies to
anycontinuous mapping of the real line to itself
The p ositive integers are usually listed in increasing order
However let us consider an alternativeenumeration that
re ects the order in which a sequence of p erio d n orbits is created For
n
instance wemight list the sequence of integers of the form as
n
where the symbol means implies In the quadratic map system
n
this ordering says that the existence of a p erio d orbit implies the
i
existence of all p erio dic orbits of p erio d for i n Wesawthis
ordering in the p erio d doubling cascade A p erio d eight orbit thus
implies the existence of b oth p erio d four and p erio d two orbits This
ordering diagram says nothing ab out the stabilityofany of these orbits
nor do es it tell us how many p erio dic orbits there are of anygiven
period
Consider the ordering of al l the integers given by
n n n n
n
with n Sarkovskii s theorem says that the ordering found in
n
equation holds in the sense of the ordering ab ove for any
CHAPTER QUADRATIC MAP
continuous map of the real line R to itself the existence of a p erio d
i orbit implies the existence of all p erio dic orbits of p erio d j where j
follows i in the ordering Sarkovskii s theorem is remarkable for its lack
of hyp otheses it assumes only that f is continuous It is of great help
in understanding the structure of one dimensional maps
In particular this ordering holds for the quadratic map For in
stance the existence of a p erio d seven orbit implies the existence of
all p erio dic orbits except a p erio d ve and a p erio d three orbit And
the existence of a single p erio d three orbit implies the existence of
p erio dic orbits of all p ossible p erio ds for the one dimensional map
Sarkovskii s theorem forces the existence of p erio d doubling cascades
in one dimensional maps It is also the basis of the famous statement
of Li and Yorke that p erio d three implies chaos where chaos lo osely
means the existence of all p ossible p erio dic orbits An elementary
pro of of Sarkovskii s theorem as well as a fuller mathematical treat
ment of maps as dynamical systems is given by Devaneyinhisbook
An Introduction to Chaotic Dynamical Systems
Sarkovskii s theorem holds only for mappings of the real line to
itself It do es not hold in the b ouncing ball system b ecause it is a map
in two dimensions It do es not hold for mappings of the circle S to
itself Still Sarkovskii s theorem is a lovely result and it do es p ointthe
way to what might b e called qualitativeuniversality that is general
statements usually top ological in nature that are exp ected to hold for
a large class of dynamical systems
Sensitive Dep endence
In section wesawhow a measurement of nite precision in the
b ouncing ball system has little predictivevalue in the long term Such
behavior is typical of motion on a chaotic attractor We called such
In section we show there exists a close connection b etween the existence
of an in nity of p erio dic orbits and the existence of a chaotic invariant set not
necessarily an attractor The term chaos in nonlinear dynamics is due to Li and
Yorke although the current usage di ers somewhat from their original de nition
see T Y Li and J A Yorke Perio d three implies chaos Am Math Monthly
SENSITIVE DEPENDENCE
behavior sensitive dep endence on initial conditions For the sp ecial
value in the quadratic map we can analyze this b ehavior in some
detail
Consider the transformation x sin applied to the quadratic
n n
map when Making use of the identity
sin sin cos
we nd
x x x
n n n
sin sin sin
n n n
sin cos
n n
sin cos
n n
sin
n
mo d
n n
This last linear di erence equation has the explicit solution
n
mo d
n
Sensitive dep endence on initial conditions is easy to see in this ex
ample when we express the initial condition as a binary numb er
X
b b b b
i
b f g
i
i
i
Now the action of equation on an initial condition is a shift
map Ateach iteration wemultiply the previous iterate bytwo in
binary which is a left shift and then apply the mo d function which
erases the integer part For example if in binary
then
mod
shift left and drop the integer part
CHAPTER QUADRATIC MAP
and we see the precision of our initial measurementevap orating b efore
our eyes
We can even quantify the amount of sensitive dep endence the system
exhibits that is the rate at which an initial error grows Assuming that
our initial condition has some small error the growth rate of the error
is
n n n n n n ln
f f e
at
If we think of n as time then the previous equation is of the form e
with a ln In this example the error grows at a constant exponential
rate of ln The exp onential growth rate of an initial error is the
de ning characteristic of motion on a chaotic attractor This rate of
growth is called the Lyapunov exponent A strictly p ositiveLyapunov
exp onent suchaswejustfound isanindicatorofchaotic motion
The Lyapunov exp onentisnever strictly p ositive for a stable p erio dic
motion
Fully Develop ed Chaos
The global dynamics of the quadratic map are well understo o d for
namely almost all orbits b eginning on the unit interval
are asymptotic to a p erio d one xed p oint We will next show that
the orbit structure is also well understo o d for This is known as
the hyp erb olic regime This parameter regime is fully develop ed in
the sense that all of the p ossible p erio dic orbits exist and they are all
unstable No chaotic attractor exists in this parameter regime but
rather a chaotic repel ler Almost all initial conditions eventually leave
or are rep elled from the unit interval However a small set remains
This remaining invariant set is an example of a fractal
The analysis found in this b o ok is based substantially on sections
to of Devaney s An Introduction to Chaotic Dynamical Systems
For a well illustrated exploration of the Lyapunov exp onent in the quadratic
map system see A K Dewdney Leaping into Lyapunov space Sci Am
pp
Technically the system is structurally stable See section of Devaney s
b o ok for more details Hyp erb olicity and structural stability usually go hand in hand
FULLY DEVELOPED CHAOS
This section is more advanced mathematically than previous sections
The reader should consult Devaney s b o ok for a complete treatment
Section contains a more pragmatic description of the symb olic
dynamics of the quadratic map and can b e read indep endently of the
current section
Hyp erb olic Invariant Sets
We b egin with some de nitions
De nition A set or region is said to b e invariant under the map f
n
if for any x wehave f x for all n
The simplest example of an invariant set is the collection of p oints
forming a p erio dic orbit But as we will see shortly there are more
complex examples such as strange invariant sets which are candidates
for chaotic attractors or rep ellers
De nition For mappings of R R a set R is a rep elling
resp attracting hyperbolic set for f if is closed b ounded and
n
invariantunderf and there exists an N suchthatj f x j
resp for all n N and all x
This de nition says that none of the derivatives of p oints in the
invariant set are exactly equal to one A simple example of a hyp erb olic
invariant set is a p erio dic orbit that is either rep elling or attracting but
not neutral In higher dimensions a similar de nition of hyp erb olicity
holds namely all the p oints in the invariantsetaresaddles
The existence of b oth a simple p erio dic regime and a complicated
fully develop ed chaotic yet well understo o d hyp erb olic regime turns
out to b e quite common in low dimensional nonlinear systems In Chap
ter we will showhow information ab out the hyp erb olic regime which
we can often analyze in detail using symb olic dynamics can b e ex
ploited to determine useful physical information ab out a nonlinear sys
tem
In examining the dynamics of the quadratic map for we
pro ceed in two steps rst we examine the invariant set and second
we describ e how orbits meander on this invariant set The set itself is
CHAPTER QUADRATIC MAP
Figure Quadratic map for Generated by the Quadratic
Map program
a fractal Cantor set and to describ e the dynamics on this fractal
set we employ the metho d of symb olic dynamics
Since f for there exists an op en interval centered at
with p oints that leave the unit interval after one iteration never to
return Call this op en set A see Figure These are the p oints in
A whose image under f is greater than one On the second iteration
more p oints leave the unit interval In fact these are the p oints that
get mapp ed to A after the rst iteration A fx I jf x A g
i n
Inductively de ne A fx I jf x I for i n but f x I g
n
that is A consists of all p oints that escap e from I at the n st
n
iteration Clearly the invariant limit set call it consists of all the
remaining p oints
I A
n
n
n
What do es lo ok like First note that A consists of disjoint n
FULLY DEVELOPED CHAOS
S S
n
op en intervals so I A A is disjoint closed in
n n
n
tervals Second f monotonically maps each of these intervals onto
n n
I The graph of f is a p olynomial with humps The maximal
sections of the humps are the collection of intervals A that get mapp ed
n
out of I but more imp ortantly this p olynomial intersects the y x
n n
line times Thus has p erio dic p oints in I
n
The set is a Cantor set if is a closed totally disconnected
p erfect subset A set is disconnected if it containsnointervals a set is
p erfect if every p oint is a limit p oint It is not to o hard to showthat
the invariant set de ned by equation is a Cantor set Thus
we see that the invariant limit set arising from the quadratic map for
is a fractal Cantor set with a countable in nity of p erio dic orbits
Symb olic Dynamics
Our next goal is to unravel the dynamics on In b eginning this task
it is useful to think how the unit interval gets stretched and folded with
each iteration The transformation of the unit interval under the rst
three iterations for is illustrated in Figure This diagram
shows that the essential ingredients that go into making a chaotic limit
set are stretching and folding The technique of symb olic dynamics is
a bookkeeping pro cedure that allows us to systematically follow this
stretching and folding pro cess For one dimensional maps the complete
symb olic theory is also known as kneading theory
We b egin by de ning a symb ol space for symb olic dynamics Let
fs s s s js or g is known as the sequencespace
j
on the symb ols and We sometimes use the symbols L Left and
R Right to denote the symb ols and see Figure If we de ne
the distance b etween two sequences s and t by
X
js t j
i i
d s t
i
i
then is a metric space The metric d s t induces a top ology on
so wehave a notion of op en and closed sets in For instance
if s andt then the metric d s t
CHAPTER QUADRATIC MAP
Figure Keeping track of the stretching and folding of the quadratic
map with symb olic dynamics
FULLY DEVELOPED CHAOS
A dynamic on the space is given by the shift map
de ned by s s s s s s That is the shift map drops the
rst entry and moves all the other symb ols one place to the left The
n
shift map is continuous Brie y forany pick n suchthat
n
and let Then the usual pro of go es through when we
use the metric given by equation
What do the orbits in lo ok like Perio dic p oints are identi ed
with exactly rep eating sequences s s s s s For
n n
instance there are two distinct p erio d one orbits given by
and The p erio d two orbit takes the form
and and one of the p erio d three orbits lo oks like
and and so on Evi
n
dently there are p erio dic p oints of p erio d n although some of these
p oints are of a lower p erio d But there is more The p erio dic p oints
are dense in that is any nonp erio dic p oint can b e represented as
the limit of some p erio dic sequence Moreover the nonp erio dic p oints
greatly outnumber the periodic points
What do es this have to do with the quadratic map or more exactly
the map f restricted to the invariant set Wenowshow that it is the
same map and thus to understand the orbit structure and dynamics
of f on we need only understand the shift map on the space of
two symb ols We can get a rough idea of the b ehavior of an orbit
bykeeping track of whether it falls to the left L or or right Ror
at the nth iteration See Figure for a picture of this partition
That is the symb ols and tell us the fold which the orbit lies on at
the nth iteration
Accordingly de ne the itinerary of x as the sequence S x s s s
j j
where s iff x ands iff x Thus the
j j
itinerary of x is an in nite sequence of s and s it lives in Fur
ther we think of S as a map from to If then it can b e
shown that S is a homeomorphism a map is a homeomor
phism if it is a bijection and b oth f and f are continuous This
last result says that the twosets and are the same To showthe
equivalence b etween the dynamics of f on and on we need the
following theorem which is quoted from Devaney
p
then S is a homeomorphism and Theorem If
CHAPTER QUADRATIC MAP
S f S
Proof See section in Devaney s b o ok This theorem holds for
all but the pro of is more subtle
As we show in the next section the essential idea in this pro of is
to keep track of the preimages of p oints not mapp ed out of the unit
interval The symb olic dynamics of the invariant set gives us a wayto
uniquely name the orbits in the quadratic map that do not run o to
in nity In particular the itinerary of an orbit allows us to name and
to nd the relative lo cation of all the p erio dic p oints in the quadratic
map Symb olic dynamics is p owerful b ecause it is easy to keep trackof
the orbits in the symb ol space It is next to imp ossible to do this using
only the quadratic map since it would involve solving p olynomials of
arbitrarily high order
Top ological Conjugacy
This last example suggests the following notion of equivalence of dy
namical systems whichwas originally put forth by Smale and is
fundamental to dynamical systems theory
De nition Let f A A and g B B be two maps The
functions f and g are said to b e topological ly conjugate if there exists
a homeomorphism h A B suchthat h f g h
The homeomorphism is called a top ological conjugacy and is more
commonly de ned by simply stating that the following diagram com
mutes
f
A A
h h
g
B B
p
Using the theorem of the previous section weknowthatif
then f the quadratic map is top ologically conjugate to the shift
FULLY DEVELOPED CHAOS
map Top ologically conjugate systems are the same system insofar as
there is a one to one corresp ondence b etween the orbits of each system
Sometimes this is to o restrictiveandwe only require that the mapping
between orbits b e many to one In this latter case wesay the two
dynamical systems are semiconjugate
In nonlinear dynamics it is often advantageous to establish a conju
gacy or a semiconjugacy b etween the dynamical system in question and
the dynamics on some symb ol space The prop erties of the dynamical
system are usually easy to see in the symb ol space and by the conju
gacy or semiconjugacy these prop erties must also exist in the original
dynamical system For instance the following prop erties are easy to
showin and must also hold in namely
The cardinality of the set of p erio dic p oints often written
n
as Per f is
n
Per f is dense in
f has a dense orbit in
Although there is no universally accepted de nition of chaos most
de nitions incorp orate some notion of sensitive dep endence on initial
conditions Our notions of top ological conjugacy and symb olic dynam
ics give us a promising way to analyze chaotic b ehavior in a sp eci c
dynamical system
In the context of one dimensional maps wesay that a map f I
I p ossesses sensitive dep endence on initial conditions if there exists a
such that for any x I and any neighborhood N of x there exist
n n
y N and n such that jf x f y j Thissays that small
errors due either to measurement or round o errors b ecome magni ed
up on iteration they cannot b e ignored
Let S denote the unit circle Here we will think of the members of
S as b eing normalized to the range A simple example of a map
that is chaotic in the ab ove sense is given by g S S de ned by
g Aswesaw in section when is written in base two
g is simply a shift map on the unit circle In ergo dic theory the ab ove
shift map is known as a Bernoulli pro cess If we think of eachsymbol
asaTail T and eachsymb ol as a Head H then the ab ove shift
CHAPTER QUADRATIC MAP
map is top ologically conjugate to a coin toss our intuitive mo del of
a random pro cess Each shift represents a toss of the coin Wenow
show that the shift map is essentially the same as the quadratic map
for that is the quadratic map a fully deterministic pro cess can
b e as random as a coin toss
If f x x x then the limit set is the whole unit inter
val I since the maximum f that is the map is
strictly onto the map is measure preserving and is roughly analogous
to a Hamiltonian system that conserves energy Tocontinue with
the analysis de ne h S by h cos Also de ne
q x x Then
h g cos
cos
q h
so h conjugates g and q Note however that h is two to one at most
points so that weonlyhave a semiconjugacy To go further if we de ne
t then f h h q Then h by h t
h h h is a top ological semiconjugacy b etween g and f wehave
established the semiconjugacy b etween the chaotic linear circle map
and the quadratic map when The reader is invited to work
through a few examples to see how the orbits of the quadratic map the
linear circle map and a coin toss can all b e mapp ed onto one another
Symb olic Co ordinates
In the previous section we showed that when the dynamics of
the quadratic map restricted to the invariant set are the same as
those given by the shift map on the sequence space on two symb ols
We established this corresp ondence by partitioning the unit inter
val into twohalves ab out the maximum p oint of the quadratic map
x The left half of the unit interval is lab eled while the right half
is lab eled as illustrated in Figure Toany orbit of the quadratic
n
map f x we assign a sequence of symb ols s s s s for ex
ample called the itinerary orsymb olic future of the orbit
SYMBOLIC COORDINATES
Each s represents the half of the unit interval in which the ith iteration
i
of the map falls
In part the theorem of section says that knowing an orbit s
initial condition is exactly equivalenttoknowing an orbit s itinerary
Indeed if we imagine that the itinerary is simply an expression for some
binary numb er then p erhaps the corresp ondence is not so surprising
n
That is the mapping f takes some initial co ordinate number x and
translates it to a binary number s s constructed from the
symb olic future which can b e thought of as a symb olic co ordinate
From a practical p oint of view the renaming scheme describ ed by
symb olic dynamics is very useful in at least twoways
Symb olic dynamics provides a go o d way to lab el all the p erio dic
orbits
The symb olic itinerary of an orbit provides the lo cation of the
orbit in phase space to any desired degree of resolution
We will explain these two p oints further and in the pro cess show the
corresp ondence b etween and x
In practical applications we shall b e most concerned with keeping
track of the p erio dic orbits Symb olic itineraries of p erio dic orbits are
rep eating nite strings which can b e written in various forms suchas
s s s s s s s s s s s s
n n n n
To see the usefulness of the symb olic description let us consider the
following problem for nd the approximate lo cation of all the
p erio dic orbits in the quadratic map
What s in a name Lo cation
As discussed in section the exact lo cation of a p erio d n orbit is
n
determined by the ro ots of the xed p oint equation f x x This
naive metho d of lo cating the p erio dic p oints is impractical in general
b ecause it requires nding the ro ots of an arbitrarily high order p oly
nomial Wenowshow that the problem is easy to solve using symb olic
dynamics if we ask not for the exact lo cation but only for the lo cation
relative to all the other p erio dic orbits
CHAPTER QUADRATIC MAP
If then there exists an interval centered ab out x for
which f x Call this interval A see Figure Clearly no
p erio dic orbit exists in A since all p oints in A leave the unit interval
I at the rst iteration and thereafter escap e to As we argued
in section the p erio dic p oints must b e part of the invariant set
those p oints that are never mapp ed into A
As shown in Figure the p oints in the invariant set can b e
constructed by considering the preimages of the unit interval found
pro duces two The rst iteration of f from the inverse map f
disjointintervals
f I
I I
which are lab eled I the left interval and I the rightinterval As
indicated by the arrows in Figure I preserves orientation while
I reverses orientation The orientation of the interval is simply deter
mined by the slop e derivative of f x
f x if x preserves orientation
f x if x reverses orientation
We view f I as a rst level approximation to the invariantset
In particular f I gives us a very rough idea as to the lo cation of
b oth p erio d one orbits one of which is lo cated somewhere in I while
the other is lo cated somewhere in I
To further re ne the lo cation of these p erio dic orbits consider the
application of f to b oth I and I
I I f f
I I I I
The second iteration gives rise to four disjointintervals Two of these
contain the distinct p erio d one orbits and the remaining twointervals
contain the p erio d two orbit
I I f I
SYMBOLIC COORDINATES
Figure Symb olic co ordinates and the alternating binary tree
CHAPTER QUADRATIC MAP
I I f I
f I I I
f I I I
n
In general we can de ne disjointintervals at the nth level of re ne
mentby
n
I f I I I f
s s s s s s
n n
With each new re nement we hone in closer and closer to the p erio dic
orbits
The one to one corresp ondence b etween x and s is easy to see ge
ometrically by observing that as n
I
s s s
n
n
forms an in nite intersection of nested nonempty closed intervals that
converges to a unique p oint in the unit interval The invariant limit
set is the collection of all such limit p oints and the p erio dic p oints are
all those limit p oints indexed by p erio dic symb olic strings
Alternating Binary Tree
Wemust keep trackoftwo pieces of information to nd the lo cation of
the orbits at the nth level the relative lo cation of the interval I
s s s
n
and its orientation A very convenientway to enco de b oth pieces of data
is through the construction of a binary tree that keeps track of all the
The quadratic map intervals generated by the inverse function f
gives rise to the alternating binary tree illustrated in Figure b
n
The nth level of the alternating binary tree has no des which are
lab eled from left to rightby the sequence
n
z
nth level
A partition of phase space that generates a one to one corresp ondence b etween
p oints in the limit set and p oints in the original phase space is known in ergo dic
theory as a generating partition Physicists lo osely call such a generating partition a go o d partition of phase space
SYMBOLIC COORDINATES
This sequence starts at the left with a zero It is followed byapairof
n
ones then a pair of zeros and so on until digits are written down
To form the alternating binary tree we construct the ab ove list of s
and s from level one to level n and then draw in the pair of lines from
each i st level no de to the adjacent no des at the ith level
Now to nd the symb olic name for the interval at the nth level
I we start at the topmost no de s and follow the path down
s s s
n
the alternating binary tree to the nth level reading o the appropriate
symb ol name at each level along the way By construction wesee
that the symb olic name read o at the nth level of the tree mimics the
lo cation of the interval containing a p erio d n orbit
More formally weidentify the set of rep eating sequences of p erio d
n in with the set of nite strings s s s Let s s s
n n
n n
denote the fraction b etween and giving the order from
left to right generated by the alternating binary tree Further let
n
N s s s denote the integer p osition b etween and
n
and let B denote N in binary form It is not to o di cult to showthat
B s s s b b b where b or and
n n i
b b b
n
s s s
n
n
n n
N s s s b b b
n n
i
X
b s mo d
i j
j
An application of the ordering relation can b e read directly o of Figure
b for n and is presented in Table As exp ected the left
most orbit is the string which corresp onds to the p erio d one orbit
at the origin x Less obvious is the p osition of the other p erio d one
orbit x which o ccupies the fth p osition at the third level
The itinerary of a p erio dic orbit is generated by a shift on the re
p eating string s s s
n
s s s s s s s
n n
In this case the shift is equivalent to a cyclic p ermutation of the sym
b olic string For instance there are two p erio d three orbits shown in
CHAPTER QUADRATIC MAP
N B s s s
x position binary x p osition symb olic name
Table Symb olic co ordinate for n from the alternating binary
tree
Table their itineraries and p ositions are
s s s
N
and
s s s
N
So the itinerary of a p erio dic orbit is generated by cyclic p ermutations
of the symb olic name
Top ological Entropy
To name a p erio dic orbit we need only cho ose one of its cyclic p er
mutations The numb er of distinct p erio dic orbits grows rapidly with
the length of the p erio d The symb olic names for all p erio dic orbits
up to p erio d eight are presented in Table A simple indicator of
the complexity of a dynamical system is its topological entropy Inthe
one dimensional setting the top ological entropy whichwe denote by
h isameasureofthegrowth of the numb er of p erio dic cycles as a
SYMBOLIC COORDINATES
Table Symb olic names for all p erio dic orbits up to p erio d eight
o ccurring in the quadratic map for All names related by a cyclic
permutation are equivalent
function of the symb ol string length p erio d
ln N
n
h lim
n
n
where N is the numb er of distinct p erio dic orbits of length n For
n
n
instance for the fully develop ed quadratic map N is of order so
n
h ln
The top ological entropy is zero in the quadratic map for anyvalue of
below the accumulation p oint of the rst p erio d doubling cascade
b ecause N is of order n in this regime The top ological entropy
n
isacontinuous monotonically increasing function b etween these two
parameter values The top ological entropy increases as p erio dic orbits
are b orn by di erent bifurcation mechanisms A strictly p ositivevalue
for the top ological entropy is sometimes taken as an indicator for the
amount of top ological chaos
In addition to its theoretical imp ortance symb olic dynamics will
also b e useful exp erimentally It will help us to lo cate and organize
the p erio dic orbit structure arising in real exp eriments In Chapter
we will showhow p erio dic orbits can b e extracted and identi ed
CHAPTER QUADRATIC MAP
from exp erimental data We will further describ e how to construct
a p erio dic orbit s symb olic name directly from exp eriments and how
to compare this with the symb olic name found from a mo del suchas
the quadratic map Reference describ es an additional re nementof
symb olic dynamics called kneading theory which is useful for analyzing
nonhyp erb olic parameter regions such as o ccur in the quadratic map
for
Notice that the ordering relation describ ed by the alternating bi
nary tree b etween the p erio dic orbits do es not change for any A
simple observation whichwillnevertheless b e very imp ortant from an
exp erimental viewp oint is the following this ordering relation which
is easy to calculate in the hyperbolic regime is often maintainedinthe
nonhyperbolic regime This is the case for instance in the quadratic
map for all This observation is useful exp erimentally b ecause it
will giveusaway to name and lo cate p erio dic orbits in an exp erimen
tal system at parameter values where a nonhyp erb olic strange attractor
exists That is we can name and identify p erio dic orbits in a hyp er
b olic regime where the system can b e analyzed analytically and then
carry over the symb olic name for the p erio dic orbit from the hyp erb olic
regime to the nonhyp erb olic regime where the system is more di cult
to study rigorously Symb olic dynamics and p erio dic orbits will b e our
breach through whichwemay attempt to p enetrate an area hitherto
deemed inaccessible
The reader might notice that our symb olic description of the
quadratic map used very little that was sp eci c to this map The
same description holds in fact for any single hump ed unimodal map
of the interval Indeed the top ological techniques we describ ed here
in terms of binary trees extend naturally to k symb ols on k ary trees
when a map with manyhumps is encountered
This concludes our intro duction to the quadratic map There are
still manymysteries in this simple map that wehavenotyet b egun
to explore such as the organization of the p erio dic window structure
but at least wecannowcontinue our journey into nonlinear lands with
words and pictures to describ e what we might see
Usage of Mathematica
Usage of Mathematica
In this section we illustrate how Mathematica can b e used for re
search or course work in nonlinear dynamics Mathematica is a com
plete system for doing symb olic graphical and numerical manipula
tions on symb ols and data and is commonly available on a wide range
of machines from micro computers s and Macs to mainframes and
sup ercomputers
Mathematica is strong in two and three dimensional graphics and
where appropriate wewould encourage its use in a nonlinear dynamics
course It can serve at least three imp ortant functions a means of
generating complex graphical representations of data from exp eriments
or simulations a metho d for double checking complex algebraic
manipulations rst done by hand and a general system for writ
ing b oth numerical and symb olic routines for the analysis of nonlinear
equations
What follows is text from a typical Mathematica session typ eset
for legibility used to pro duce some of the graphics and to double check
some of the algebraic results presented in this chapter Of course a
real Mathematica session would not b e so heavily commented
This is a Mathematica comment statement Mathematica ignores
everything between star parentheses
To try out this Mathematica session yourself type everything in
b old that is not enclosed in the comment statements The following
line is Mathematica s answer typed in italic Mathematica s output
from graphical commands are not printed here but are left for the
reader to discover
This notebook is written on a Macintosh nbt
In this notebook we will analytically solve for the period one and
period two orbits in the quadratic map and plot their locations as a
function of the control parameter lambda
Mathematica is a trademark of Wolfram Research Inc For a brief intro duction
to Mathematica see S Wolfram Mathematica a system for doing mathematics by
computer Addison Wesley New York pp
Quadratic Map
First we define the quadratic function with the Mathematica
Function fx y g f x y command which takes two
arguments the first of which is a list of variables and the second
of which is the function The basic data type in Mathematica
is a list and all lists are enclosed between braces
fx x x g Note that all arguments and functions in
Mathematica are enclosed in square brackets f which differs from
the standard mathematical notation of parentheses f This is in
part because square brackets are easier to reach on the keyboard
Also note that in defining a variable one must always put a space
around it so xy is equal to a single variable named xy while
x y with a space between is equal to two variables x and y
To evaluate a Mathematica expression tap the enter key not the
return key Now to our first Mathematica command
f Function flambda xg lamb da x x
Function flambda xg lambda x x
Mathematica should respond by saying Out which tells us that
Mathematica successfully processed the first command and has put the
result in the variable Out as well as the variable we created f
To evaluate the quadratic map we now can feed f two arguments
the first of which is lambda and the second of which is x
f
Mathematica should respond with the function Out which
contains the quadratic map evaluated at lambda and x To
evaluate a list of values for x we could let the x variable be a
list i e a series of numbers enclosed in braces fx x x g
To try this command type
f f g
f g
Usage of Mathematica
To plot the quadratic map we use the Mathematica plot command
Plot f fx xmin xmaxg For instance a plot of the quadratic map
for lambda is given by
Plot f x fx g
It is as easy as pie to take a composite function in Mathematica
just type f f x so to plot f f x for the quadratic map we
simply type
Plot f f x fx g
Now let s find the locations of the period one orbits given by
the roots of f x x To find the roots we use the Mathematica
command Solve eqns vars Notice that we are going to rename the
parameter lambda to a just to save some space when printing the
answer The double equals in Mathematica is equivalent to the
single equal of mathematics
Solve f a x x x
ffx a g fx gg
As expected Mathematica finds two roots Let s make a function
out of the first root so that we can plot it later using Plot To do
this we need the following sequence of somewhat cryptic commands
r
fx a g
The roots are saved in a list of two items To pull out the
first item we used the command a Mathematica variable that always
holds the value of the last expression In this case it holds the
list of two roots and the double square bracket notation
tells Mathematica we want the first item in the list of two items
Now we must pull out the last part of the expression a with
the replacement command Replace expr rules
Quadratic Map
x Replace x r
a
To plot the location of the period one orbit we just use the Plot
command again
Plot x fa g
To find the location of the period two orbit
we solve for f f x x
Solve f a f a x x x
ffx g fx a g
fx a a a a g
fx a a a a gg
We find four roots as expected Before proceeding further it s
a good idea to try and simplify the algebra for the last two new
roots by applying the Simplify command to the last expression
rt Simplify
ffx g fx a g
fx a a a g
fx a a a gg
And we can now pull out the positive and negative roots of the
period two orbit
x plus Replace x rt
x minus Replace x rt
a a a
As a last step we can plot the location of the period one orbit
and both branches of the period two orbit by making a list of
functions to be plotted
References and Notes
Plot fx x plus x minusg fa g
This is how we originally plotted the orbit stability diagram
in the text Mathematica can go on to find higher order periodic
orbits by numerically finding the roots of the nth composite of f
if no exact solution exists
References and Notes
There are several excellent reviews of the quadratic map Some of the oldest
are still the b est and all of the following are quite accessible to undergraduates
E N Lorenz The problem of deducing the climate from the governing equa
tions Tellus E N Lorenz Deterministic nonp erio dic ow J
Atmos Sci R M May Simple mathematical mo dels with
very complicated dynamics Nature M J Feigenbaum
Universal b ehavior in nonlinear systems Los Alamos Science
These last two articles are reprinted in the b o ok P Cvitanovi c ed Universal
ity in Chaos Adam Hidgler Ltd Bristol All four are mo dels of go o d
exp ository writing A simple circuit providing an analog simulation of the
quadratic map suitable for an undergraduate lab is describ ed by T Mishina
T Kohmoto and T Hashi Simple electronic circuit for the demonstration of
chaotic phenomena Am J Phys
A go o d review of the dynamics of the quadratic map from a dynamical systems
p ersp ectiveisgiven byR L Devaney Dynamics of simple maps in Chaos and
fractals The mathematics behind the computer graphics Pro c Symp Applied
Math edited by R L Devaney and L Keen AMS Rho de Island
For an elementary pro of see E A Jackson Perspectives in nonlinear dynamics
Vol Cambridge University Press New York pp
Counting the numb er of p erio dic orbits of p erio d n isavery prettycombina
torial problem See Hao B L Elementary symbolic dynamics and chaos in
dissipative systems World Scienti c New Jersey pp An up
dated version for some results in this b o ok can b e found in Zheng W M and
Hao B L Applied symb olic dynamics in Experimental study and characteri
zation of chaos editedby Hao B L World Scienti c New Jersey Also
see the original analysis published in the article by M Metrop olis M L Stein
and P R Stein On the nite limit sets for transformations of the unit interval
J Comb Theory Also reprinted in Cvitanovi c reference
Quadratic Map
For a discussion concerning the existence of a single stable attractor in the
quadratic map and its relation to the Schwarzian derivative see Jackson ref
erence pp and App endix D pp
A more complete mathematical account of lo cal bifurcation theory is presented
by S N Rasband Chaotic dynamics of nonlinear systems John Wiley Sons
New York pp and pp Chapter deals with universality
theory from the quadratic map
G Io oss and D D Joseph Elementary stability and bifurcation theory Springer
Verlag New York
For more ab out this tale see the chapter Universality of J Gleick Chaos
Making a new science Viking New York
See the intro duction of P Cvitanovi c ed Universality in Chaos Adam Hidgler
Ltd Bristol
R L Devaney Anintroduction to chaotic dynamical systems second edition
Addison Wesley New York Section covers Sarkovskii s Theo
rem and section covers kneading theory Another elementary pro of of
Sarkovskii s Theorem can b e found in H Kaplan A carto on assisted pro of of
Sarkovskii s Theorem Am J Phys
K Falconer Fractal geometry John Wiley Sons New York
S Smale The mathematics of time Essays on dynamical systems economic
processes and related topics Springer Verlag New York
P Cvitanovi c G H Gunaratne and I Pro caccia Top ological and metric
prop erties of H enon typ e strange attractors Phys Rev A
Some pap ers providing a hands on approach to kneading theory include P
Grassb erger On symb olic dynamics of one hump ed maps of the interval Z
Naturforsch a J P Allouche and M Cosnard It erations
de fonctions unimo dales et suites engendr ees par automates C R Acad Sc
Paris S erie I and reference Also see section of
Devaney s b o ok reference for a nice mathematical account of kneading
classic reference in kneading theory is J Milnor and W Thurston theory The
On iterated maps of the interval Lect Notes in Math in Dynamical Sys
tems Pro ceedings University of Maryland edited by J C Alexander
Springer Verlag Berlin pp The early mathematical con
tributions of Fatou Julia and Myrb erg to the dynamics of maps symb olic
dynamics and kneading theory are emphasized by C Mira Chaotic dynamics
From the one dimensional endomorphism to the two dimensional di eomor
phism World Scienti c New Jersey
Problems
H Poincar e Les m ethodes nouvel les de la m ecanique c eleste Vol Gauthier
Villars Paris reprinted byDover English translation New
metho ds of celestial mechanics NASA Technical Translations See Vol
section for the quote
See sections and of Hao B L reference for some results on
p erio dic windows in the quadratic map
Problems
Problems for section
For the quadratic map eq show that the interval is a
trapping region for all x in the unit interval and all
Use the transformation
x y
to show that the quadratic map eq can b e written as
y y y
n
n
or using a di erent x transformation as
z z z
n
n
Sp ecify the ranges to which x and are restricted under these transforma
tions
Read the Tellus article by Lorenz mentioned in reference
Section
Write a program to calculate the iterates of the quadratic map
Section
For f x x x and x calculate f x by the graphical metho d
describ ed in section
Quadratic Map
Showby graphical analysis that if x and in the quadratic map
n
then as n f x Further show that if then the
xed p oint at the origin is an attractor
Section
Find all the attractors and basins of attraction for the map f x mx where
m is a constant
The tent map see sections of Rasband reference is de ned by
x if x
jx j
x if x
a Sketch the graph of the tent map for Why is it called the tent
map
b Show that the xed p oints for the tent map are
x and for x
c Showthat x is always rep elling and that x is attracting when
d For a one dimensional map the Lyapunov exp onent is de ned by
d
n
ln f x x lim
n
n dx
x x
Show that for theLyapunov exp onent for the tentmapis ln
Hint For the tent map use the chain rule of di erentiation to showthat
n
X
ln jf x j x lim
i
n
n
i
Determine the lo cal stability of orbits with f x and f x
using graphical analysis as in Figure
Section
Determine the parameter value s for which the quadratic map intersects the
line y x just once
Consider a p erio d two orbit of a one dimensional map
x f f x f x
s s s
Problems
a Use the chain rule for di erentiation to show that f x f x
b Show that the p erio d two orbit is stable if jf x f x j and
unstable if jf x f x j
For use Mathematica to nd the lo cations of b oth p erio d three orbits
in the quadratic map
Consider a p cycle of a one dimensional map If this p cycle is p erio dic then
p
it is a xed p ointoff Letx x x represent one orbit of p erio d p
p
Show that this p erio dic orbit is stable if
p
Y
jf x j
j
j
Hint Showby the chain rule for di erentiation that for any orbit not nec
essarily p erio dic
p
df
f x f x f x
p
dx
x x
Now assume the orbit is p erio dic Then for anytwopoints of a p erio d p orbit
p p
x and x note that f x f x
i j i j
Consider a seed near a p erio d two orbit x x with slop e near to
n
f x Showby graphical analysis that f x ips back and forth
between the twopoints on the p erio d two orbit Showby graphical construc
tion that this p erio d two orbit is stable if j f x j and unstable if
j f x j
n n
For show that the quadratic map f x intersects the y x line
times but only some of these intersection p oints b elong to new p erio dic orbits
of p erio d n For n to build a table showing the numb er of di erent
orbits of f of p erio d n
n
Prove see Prob that for prime n n is an integer this result
is a sp ecial case of the Simple Theorem of Fermat
a Derive the equation of the graph shown in Figure
b Verify that the twointersection p oints shown in the gure o ccur at and
p
Equation follows directly from equation and the chain rule dis
cussion in Problem Derive it using only equations and
Quadratic Map
Section
Write a program to generate a plot of the bifurcation diagram for the quadratic
map
Section
Show that the p erio d two orbit in the quadratic map loses stabilityat
p
i e f x at this value of
Show that the equispaced orbits of the b ouncing ball system see Prob
are b orn in a saddle no de bifurcation Note that equation gives
two impact phases for each n For n andforn in equation
which orbit is a saddle near birth and which orbit is a no de What is the
impact phase of the saddle What is the impact phase of the no de
Use Mathematica or another computer program to show that a pair of p erio d
three orbits are b orn in the quadratic map by a tangent bifurcation at
p
n
Show that the absolute value of the slop e of f evaluated at all p oints of a
period n orbit have the same value at a bifurcation p oint
In Figure the two sets of triangles op en white interior and closed
blackinterior representtwo di erent p erio d three orbits Show that the op en
triangles represent an unstable orbit
Section
Using Table and equations and estimate c and
Use the bifurcation diagram of the quadratic map Fig and a ruler
to measure the rst few p erio d doubling bifurcation values Itmaybe
n
helpful to use a photo copying machine to expand the gure b efore doing the
measurements Based on these measurements estimate Feigenbaum s delta
with equation Do the same thing for the bifurcation diagram for the
b ouncing ball system found in Figure How do these twovalues compare
Section
Find a one dimensional circle map g S S for whichSarkovskii s ordering
do es not hold Hint Find a map that has no p erio d one solutions by using a
discontinuous map
Consider the p erio dic orbits of p erio ds and Order
these p oints according to Sarkovskii s ordering equation Show that
Sarkovskii s ordering uniquely orders all the p ositiveintegers
Problems
Section
For determine the interval A asshown in Figure as a function of
Verify that p oints in this interval leave the unit interval and never return
a Using the metric of equation calculate the distance b etween the
twoperiodtwo p oints and
b Create a table showing the distances b etween all six p erio d three p oints
and
Establish an isomorphism b etween the unit interval and the sequence space
of section Hint See Devaney section
a De ne f R R by f x mx b and de ne g R R by g x
mx nb where m b andn R Showthat f and g are top ologically
conjugate
b De ne f S S by f x x De ne g by
g x x Show that f and g are top ologically semiconjugate
c Find a set of functions f g andh that satis es the de nition of top ological
conjugacy
Section
Construct the binary tree up to the fourth level where the nth level is de ned
by the rule
n
z
nth level
a Construct the sixteen symb olic co ordinates s s s s at the fourth level
of this binary tree and show that the ordering from left to rightatthe nth
level is given by
n n
N s s s s s s
n n
Why is it called the binary tree
b Show that the fractional ordering is given by
s s s
n
s s s
n
n
c Give an example of a one dimensional map not necessarily continuous
on the unit interval giving rise to the binary tree
Quadratic Map
Construct the alternating binary tree up to the fourth level and calculate the
symb olic co ordinate and p osition of each of the sixteen p oints Present
this information in a table
Show that the order on the x axis of twopoints x and y in the quadratic
map with is determined by their itineraries fa g and fb g as follows
k k
supp ose a b a b a b and that a and b i e
k k k k
the itineraries of each initial condition are identicaluptothe k th iteration
and di er for the rst time at the k st iteration Then
k
X
x y a mo d
i
i
Hint See App endix A of reference and theorem on page of
Devaney reference
Chapter
String
Intro duction
Like a jump rop e a string tends to swing in an ellipse a fact well known
to children When holding b oth ends of a rop e or string it is di cult to
shake it so that motion is con ned to a single transverse plane Instead
of remaining con ned to planar oscillations strings app ear to prefer
elliptical or whirling motions like those found when playing jump rop e
Borrowing terminology from optics wewould say that a string prefers
circular p olarization to planar p olarization In addition to whirling
other phenomena are easily observed in forced strings including bi
furcations b etween planar and nonplanar p erio dic motions transitions
to chaotic motions sudden jumps b etween di erent p erio dic motions
hysteresis and p erio dic and ap erio dic cycling b etween large and small
vibrations
In this chapter we will b egin to explore the dynamics of an elastic
string by examining a single mo de mo del for string vibrations In the
pro cess several new typ es of nonlinear phenomena will b e discovered
including a new typ e of attractor the torus arising from quasip erio dic
motions and a new route to chaos via torus doubling We will also
showhowpower sp ectra and Poincar e sections are used in exp eriments
to identify di erenttyp es of nonlinear attractors In this waywe will
continue building the vo cabulary used in studying nonlinear phenomena
CHAPTER STRING
In addition to its intrinsic interest understanding the dynamics of
a string can also b e imp ortantformusicians instrumentmakers and
acoustical engineers For instance nonlinearity leads to the mo dulation
and complex tonal structure of sounds from a cello or guitar Whirling
motions account for the rattling heard when a string is strongly plucked
Linear theory provides the basic outline for the science of the
pro duction of musical sounds its real richness though comes from
nonlinear elements
When a string vibrates the length of the string must uctuate
These uctuations can b e along the direction of the string longitu
dinal vibrations or up and down vibrations transverse to the string
The longitudinal oscillations o ccur at ab out twice the frequency of the
transverse vibrations The mo dulation of a string s length is the essen
tial source of a string s nonlinearity and its rich dynamical b ehavior
The coupling b etween the transverse and longitudinal motions is an
example of a parametric oscil lation An oscillation is said to b e para
metric when some parameter of a system is mo dulated in this case the
string s length Linear theory predicts that a string s free transverse os
cillation frequency is indep endent of the string s vibration amplitude
Exp erimental measurements on the other hand show that the reso
nance frequency dep ends on the amplitude Thus the linear theory has
a restricted range of applicability
Think of a guitar string A string under a greater tension has a
higher pitch fundamental frequency Whenever a string vibrates it
gets stretched a little more so its pitch increases slightly as its vibration
amplitude increases
We b egin this chapter by describing the exp erimental apparatus
we ve used to study the string section In section wemodel
our exp eriment mathematically Sections to examine a sp ecial
case of string b ehavior planar motion whichgives rise to the Du ng
equation Section lo oks at the more general case nonplanar mo
tion Finally in section we present exp erimental techniques used
by nonlinear dynamicists These exp erimental metho ds are illustrated
in the string exp eriment
EXPERIMENTAL APPARATUS
Figure Schematic of the apparatus used to study the vibrations of
a wire string
Exp erimental Apparatus
An exp erimental apparatus to study the vibrations of a string can b e
constructed bymounting a wire b etween twoheavy brass anchors
As shown in Figure a screw is used to adjust the p osition of the
anchors and hence the tension in the wire string An alternating sinu
soidal current passed through the wire excites vibrations this currentis
usually supplied directly from a function generator An electromagnet
or large p ermanent magnet is placed at the wire s midp oint The inter
action b etween this magnetic eld and the magnetic eld generated by
the wire s alternating current causes a p erio dic force to b e applied at
the wire s midp oint If a nonmagnetic wire is used such as tungsten
then b oth planar and nonplanar whirling motions are easy to observe
On the other hand if a magnetic wire is used such as steel then the
motion always remains restricted to a single plane The use of a
magnetic wire intro duces an asymmetry into the system that causes
the damping rate to dep end strongly on the direction of oscillation
From the Lorentz forcelaw a wire carrying a current I in a magnetic eld of
R
strength B is acted on by a magnetic force F I B dl If the current I in
mag
the wire varies sinusoidally then so do es the force on the wire See D J Gri ths
An introduction to electrodynamics Prentice Hall Englewo o d Cli s NJ pp
CHAPTER STRING
A similar asymmetry is seen in the decay rates of violin guitar
and piano strings In these musical instruments the string runs over a
bridge which helps to hold the string in place The bridge damps the
motion of the string however the damping force applied by the bridge
is di erent in the horizontal and vertical directions The clamps
holding the string in our apparatus are designed to b e symmetric and
it is easy to check exp erimentally that the decay rates in di erent direc
tions in the absence of a magnetic eld show no signi cantvariation
Our exp erimental apparatus can b e thoughtofastheinverse of that
found in an electric guitar There a magnetic coil is used to detect the
motion of a string In our apparatus an alternating magnetic eld is
used to excite motions in a wire
The horizontal and vertical string displacements are monitored with
a pair of inexp ensive slotted optical sensors consisting of an LED light
emitting dio de and a phototransistor in a U shap ed plastic housing
Two optical detectors one for the horizontal motion and one for the
vertical motion are mounted together in a holder that is fastened to
a microp ositioner allowing exact placement of the detectors relative
to the string The detectors are typically p ositioned near the string
mounts This is b ecause the detector s sensitivity is restricted to a
small amplitude range and the string displacementisminimal close to
the string mounts As shown in Figure the string is p ositioned
to obstruct the light from the LED and hence casts a shadowonthe
surface of the phototransistor For a small range of the string displace
ments the size of this shadow is linearly prop ortional to the p osition of
the string and hence also to the output voltage from the photo detec
tor This voltage is then monitored on an oscilloscop e digitized with
a micro computer or further pro cessed electronically to construct an
exp erimental Poincar e section as describ ed in section
Care must b e taken to isolate the rig mechanically and acoustically
In our case wemounted the apparatus on a oating optical table We
also constructed a plastic cover to provide acoustical isolation The
string apparatus is small and easily ts on a desktop Typical exp eri
mental parameters are listed in Table
Most of our theoretical analysis will b e concerned with single mo de
oscillations of a string If wepluck the string near its center it tends
to oscillate in a sinusoidal manner with most of its energy at some pri
EXPERIMENTAL APPARATUS
Figure Mo dule used to detect string displacements Adapted from
Hanson
Parameter Typical exp erimental value
Length mm
Mass p er unit length g m
Diameter mm
Primary resonance kHz
Range of hysteresis Hz
Magnetic eld strength T
Current A
Maximum displacement mm
Damping
Table Parameters for the string apparatus
CHAPTER STRING
mary frequency called the fundamental A similar plucking e ect can
be achieved by exciting wire vibrations with the current and the sta
tionary magnetic eld Topluck the string we switch o the current
after we get a large amplitude string vibration going The fundamen
tal frequency is recognizable byusasthecharacteristic pitchwe hear
when the string is plucked A large amplitude resonant resp onse is
exp ected when the forcing frequency applied to a string is near to this
fundamental This is the primary resonance of the string and it is
de ned by the linear theory as
T
l
where m l is the mass p er unit length and T is the tension in the
string
The primary assumption of the linear theory is that the equilibrium
length of the string l remains unchanged as the string vibrates that
is l t l where l t is the instantaneous length In other words the
linear theory assumes that there are no longitudinal oscillations In
developing a simple nonlinear mo del for the vibrations of a string we
must b egin to takeinto account these longitudinal oscillations and the
dep endence of the string s length on the vibration amplitude
Single Mo de Mo del
A mo del of a string oscillating in its fundamental mo de is presented in
Figure and consists of a single mass fastened to the central axis bya
pair of linearly elastic springs Although the springs provide a linear
restoring force the resulting force toward the origin is nonlinear b ecause
of the geometric con guration The ends of the massless springs are
xed a distance l apart where the relaxed length of the spring is l
and the spring constantisk In the center a mass is attached that is
free to make oscillations in the x y plane centered at the origin The
motion in the two transverse directions x and y is coupled directly
For a review of the linear theory for the vibrations of a stretched string see A
P French Vibrations and waves W W Norton New York pp
SINGLE MODE MODEL
Figure Single mo de mo del for nonlinear string vibrations String
vibrations are assumed to b e in the fundamental mo de and are mea
sured in the transverse x y plane by the p olar co ordinates r a
Equilibrium length b relaxed length
and also indirectly via the longitudinal motion of the spring Both of
these coupling mechanisms are nonlinear The multimo de extension of
this single mo de mo del would consist of n masses ho oked together by
n springs
The restoring force on the mass shown in Figure is
l
p
F kr
l r
where the p osition of the mass is given by p olar co ordinates r of
the transverse plane see Prob Expanding the right hand side of
equation in a Taylor series r l we nd that
r r r
F kr l l kl
l l l
The force can b e written as
F mr
CHAPTER STRING
so
l r r
mr k l l
l l l l
Note the cubic restoring force Also note that nonlinearity dominates
when l l That is the nonlinear e ects are accentuated when the
string s tension is low
De ne
k l l
m l
and
l
K
l l l
Then from equation we get b ecause of symmetry in the angular
co ordinate the vector equation for r x t y t
r K r r
which is the equation of motion for a two dimensional conservative
cubic oscillator The b ehavior of equation dep ends critically
up on the ratio l l If l l the co e cient of the nonlinear term K
is p ositive the equilibrium p ointat r is stable and wehave a mo del
for a string vibrating primarily in its fundamental mo de On the other
hand if l l then K is negative the origin is an unstable equilibrium
point and two stable equilibrium p oints exist at approximately r l
This latter case mo dels the motions of a single mo de elastic b eam
For our purp ose we will mostly b e concerned with the case l l or
K
In general wewillwant to consider damping and forcing so equa
tion is mo di ed to read
r r K r r f t
where f t is a p erio dic forcing term and is the damping co e cient
Usually the forcing term is just a sinusoidal function applied in one
radial direction so that it takes the form f t A cos t For
The term r in equation is a typical physicist s notation meaning the dot
pro duct of the vector r r r x y r
SINGLE MODE MODEL
simplicity wehave assumed that the energy losses are linearly prop or
tional to the radial velo city of the string r We also assumed that the
ends of the string are symmetrically xed so that is a scalar In gen
eral the damping rate dep ends on the radial direction so the damping
term is a vector function This is the case for instance when a string
is strung over a bridge that breaks the symmetry of the damping term
Equation was also derived by Gough and Elliot b oth
of whom related and K to actual string parameters that arise in
exp eriments For instance Gough showed that the natural frequency
is given by
c
l
and the strength of the nonlinearityis
K
l
where is the longitudinal extension of a string of equilibrium length l
is the low amplitude angular frequency of free vibration and c is the
transverse wavevelo city Again we see that the nonlinearity parameter
K increases as the longitudinal extension approaches zero That
is the nonlinearity is enhanced when the longitudinal extension and
hence the tension is small Nonlinear e ects are also ampli ed when
the overall string length is shortened and they are easily observable
in common musical instruments For a viola D string with a vibration
amplitude of mm typical values of the string parameters showing
nonlinear e ects are l cm Hz mm K
mm
Equation constitutes our single mo de mo del for nonlinear string
vibrations and is the central result of this section For some calcula
tions it will b e advantageous to write equation in a dimensionless
form To this end consider the transformation
r
t s
l
which gives
s s s s g
CHAPTER STRING
where the prime denotes di erentiation with resp ect to and
f
g Kl and
l
Before we b egin a systematic investigation of the single mo de mo del
it is useful to consider the unforced linear problem f t If the
nonlinearity parameter K is zero then equation is simply a two
degree of freedom linear harmonic oscillator with damping that admits
solutions of the form
t
r X cos t Y sin t e
where X and Y are the initial amplitudes in the x and y directions
Equation is a solution of if we discard second order terms
in In the conservative limit the orbits are ellipses centered
ab out the z axis As weshow in section one e ect of the nonlin
earity is to cause these elliptical orbits to precess The tra jectories of
these precessing orbits resemble Lissa jous gures and these precessing
orbits will b e one of our rst examples of quasip erio dic motion on a
torus attractor
Planar Vibrations Du ng Equation
An external magnetic eld surrounding a magnetic wire restricts the
forced vibrations of a wire to a single plane Alternatively we could
fasten the ends of the wire in sucha way as to constrain the motion
to planar oscillations In either case the nonlinear equation of motion
governing the single mo de planar vibrations of a string is the Du ng
equation
x x x Kx A cos t
where equation is calculated from equation by assuming
that the string s motion is con ned to the x z plane in Figure The
forcing term in equation is assumed to b e a p erio dic excitation of
the form
f t A cos t
PLANAR VIBRATIONS DUFFING EQUATION
where the constant A is the forcing amplitude and is the forcing
frequency The literature studying the Du ng equation is extensive
anditiswell known that the solutions to equation are already
complicated enough to exhibit multiple p erio dic solutions quasip eri
o dic orbits and chaos A go o d guide to the nonchaotic prop erties of
the Du ng equation is the b o ok byNayfeh and Mo ok Highly
recommended as a pioneering work in nonlinear dynamics is the b o ok
byHayashi which deals almost exclusively with the Du ng equation
Equili bri um States
The rst step in analyzing any nonlinear system is the identi cation of
its equilibrium states The equilibrium states are the stationary p oints
of the system that is where the system comes to rest For a system of
di erential equations the equilibrium states are calculated by setting
all the time derivatives equal to zero in the unforced system Setting
x x and A in equation we immediately nd that
the lo cation of the equilibrium solutions is given by
x Kx
which in general has three solutions
s s
x and x x
K K
Clearly there is only one real solution if K x since the other two
solutions x are imaginary in this case If K then there are three
real solutions
To understand the stability of the stationary p oints it is useful to
recall the physical mo del that go es with equation If l l then
K see eqs and and the Du ng equation is a
simple mo del for a b eam under a compressive load As illustrated in
Figure the solutions x corresp ond to the two asymmetric stable
b eam con gurations The p osition x corresp onds to the symmetric
unstable b eam con guration a small tap on the b eam would imme
diately send it to one of the x con gurations If l l thenK
CHAPTER STRING
Figure Equilibrium states of a b eam under a compressive load
Figure Equilibrium state of a wire under tension
and the Du ng equation is a simple mo del of a string or wire under
tension so there is only one symmetric stable con guration x Fig
Unforced Phase Plane
Conservative Case
After identifying the equilibrium states our next step is to understand
the tra jectories in phase space in a few limiting cases In the unforced
conservative limit a complete account of the orbit structure is given by
integrating the equations of motion by using the chain rule in the form
dv dx d
v x x v
dt dx dt
dv
v
dx
Applying this identity to equation with and A yields
dv
v x Kx
dx
whichcanbeintegrated to give
x x
v h K
PLANAR VIBRATIONS DUFFING EQUATION
where h is the constantofintegration
The term on the left hand side of equation is prop ortional to
the kinetic energy while the term on the right hand side
x x
V x K
is prop ortional to the p otential energy Therefore the constant h is
prop ortional to the total energy of the system as illustrated in Figure
a
The phase space is a plot of the p osition x and the velo city v of all
the orbits in the system In this case the phase space is a phase plane
and in the unforced conservative limit we nd
p
h V x v
p
x x
h K
The last equation allows us to explicitly construct the integral curves a
plot of v t vs x t in the phase plane Eachintegral curve is lab eled
bya value of h and the qualitative features of the phase plane dep end
critically on the signs of and K
and K are p ositive A plot of equation If l l then b oth
for several values of h is given in Figure c If h h
then the integral curve consists of a single p oint called a center When
h h the orbits are closed b ounded simply connected curves ab out
the center Eachcurve corresp onds to a distinct p erio dic motion of the
system Going back to the string mo del again we see that the center
corresp onds to the symmetric equilibrium state of the string while the
integral curves ab out the center corresp ond to nite amplitude p erio dic
oscillations ab out this equilibrium p oint
If l l then K is negative The phase plane has three stationary
p oints This parameter regime mo dels a compressed b eam The left and
right stationary p oints x are centers but the unstable p ointat x
lab eled S in Figure b is a sadd le point b ecause it corresp onds to
a lo cal maximum of V x
Curves that pass through a saddle p oint are very imp ortantand
are called separatrices In Figure d we see that there are two
CHAPTER STRING
Figure Potential and phase space for a single mo de string a c e and b eam b d f
PLANAR VIBRATIONS DUFFING EQUATION
integral curves approaching the saddle p ointS andtwointegral curves
departing from S These separatrices separate the phase plane into
two distinct regions Eachintegral curve inside the separatrices go es
around one center and hence corresp onds to an asymmetric p erio dic
oscillation ab out either the left or the rightcenter but not b oth The
integral curves outside the separatrices go around all three stationary
p oints and corresp ond to large amplitude symmetric p erio dic orbits
Figure d Thus the separatrices act like barriers in phase space
separating motions that are qualitatively di erent
DissipativeCase
If damping is included in the system then the phase plane changes
to that shown in Figure e For the string damping destroys all
the p erio dic orbits and all the motions are damp ed oscillations that
converge to the point attractor at the origin That is if we plucka
string the sound fades away The string vibrates with a smaller and
smaller amplitude until it comes to rest Moreover the basin of attrac
tion for the p oint attractor is the entire phase plane This particular
p oint attractor is an example of a sink
The phase plane for the oscillations of a damp ed b eam is a bit more
involved as shown in Figure f The center p oints at x b ecome
p oint attractors while the stationary p ointat x is a saddle There are
two separate basins of attraction one for eachpoint attractor sink
The shaded region shows all the integral curves that head toward the
right sink Again we see the imp ortant role played by separatrices since
they separate the basins of attraction of the left and right attracting
p oints In the context of a dissipative system the separatrix naturally
divides into two parts the inset consisting of all integral curves that
approach the saddle p oint S and the outset consisting of all p oints
departing from S Formally the outset of S can b e de ned as all p oints
that approach S as time runs backwards That is we simply reverse all
the arrows in Figure f
The qualitative analysis of a dynamical system can usually b e di
vided into two tasks rst identify all the attractors and rep ellers of
the system and second analyze their resp ective insets and outsets
Attractors and rep ellers are limit sets Insets outsets and limit sets
CHAPTER STRING
are all examples of invariant sets see section Thus muchof
dynamical systems theory is concerned not simply with the analysis of
attractors but rather with the analysis of invariant sets of all kinds at
tractors rep ellers insets and outsets For the unforced damp ed b eam
the task is relatively easy There are two attracting p oints and one
saddle p oint The inset and outset of the saddle p oint spiral around
the two attracting p oints and completely determine the structure of the
basins of attraction see Figure f
Extended Phase Space
To continue with the analysis of planar string vibrations wenow turn
our attention to the forced Du ng equation in dimensionless variables
from eqs and
x x x x F cos
where F is the forcing amplitude and is the normalized forcing fre
quency
It is often useful to rewrite an nth order di erential equation as a
system of rst order equations and to recall the geometric interpreta
tion of a di erential equation as a vector eld To this end consider
the change of variable v x sothat
x v
x v R
v aut x v g
where aut x v v x x isthe autonomous or time
indep endent term of v and g F cos is the time dep endent
term of v The phase space for the forced Du ng equation is top olog
ically a plane since each dep endentvariable is just a copyofR and
the phase space is formally constructed from the Cartesian pro duct of
these two sets R R R
Avector eld is obtained when to each p oint on the phase plane
we assign a vector whose co ordinate values are equal to the di erential
system evaluated at that p oint The vector eld for the unforced un
damp ed Du ng equation is shown in Figure a This vector eld
is static time indep endent In contrast the forced Du ng equation
PLANAR VIBRATIONS DUFFING EQUATION
Figure Extended phase space for the Du ng oscillator
has a time dep endentvector eld since the value of the vector eld at
x v at is
x v v aut x v F cos
In Figure b weshowwhattheintegral curves lo ok like when plotted
in the extended phase space which is obtained byintro ducing a third
variable
z
With this variable the di erential system can b e rewritten as
x v
aut x v g z v
x v z R
z
CHAPTER STRING
By increasing the numb er of dep endentvariables byone we can for
mally change the forced time dep endent system into an autonomous
time indep endent system
Moreover since the vector eld is a p erio dic function in z itis
sensible to intro duce the further transformation
mo d
thereby making the third variable top ologically a circle S With this
transformation the forced Du ng equation b ecomes
x v
v v x x F cos
x v R S
One last reduction is p ossible in the top ology of the phase space of
the Du ng equation It is usually p ossible to nd a trapping region
that top ologically is a disk a circular subset D R In this last
instance the top ology of the phase space for the Du ng equation is
simply D S orasolid torus see Figure c
Global Cross Section
The global solution to a system of di erential equations the collection
of all integral curves is also known as a ow A ow is a one parameter
family of di eomorphisms of the phase space to itself see section
To visualize the ow in the Du ng equation imagine the extended
phase space as the solid torus illustrated in Figure Each initial
condition in the disk D at must return to D when
b ecause D is a trapping region and the variable is p erio dic That
is the region D ows back to itself An initial p ointinD lab eled
x v is carried by its integral curve back to some new p oint
lab eled x v also in D The Du ng equation satis es the
fundamental uniqueness and existence theorems in the theory of or
dinary di erential equations Hence each initial p ointinD gets
carried to a unique p ointbackin D and no twointegral curves can ever
intersect in D S
As originally observed byPoincar e this unique dep endence with resp ect to initial conditions along with the existence of some region in
PLANAR VIBRATIONS DUFFING EQUATION
Figure Phase space for the Du ng oscillator as a solid torus
phase space that is recurrent allows one to naturally asso ciate a map
to any ow The map he describ ed is now called the Poincar emap For
the Du ng equation this map is constructed from the ow as follows
of the vector eld eq by De ne a global cross section
f x v D S j g
Next de ne the Poincar e map of as
P x x v v
where x v is the next intersection with of the integral curve
emanating from x v For the Du ng equation the Poincar e map is
also known as a strob oscopic map since it samples or strob es the ow
at a xed time interval
The dynamics of the Poincar e map are often easier to study than
the dynamics in the original ow By constructing the Poincar emap
we reduce the dimension of the problem from three to two This di
mension reduction is imp ortant b oth for conceptual clarityaswell as
CHAPTER STRING
for graphical representations b oth numerical and exp erimental of the
dynamics For instance a p erio dic orbit is a closed curveinthe ow
The corresp onding p erio dic orbit in the map is a collection of p oints
in the map so the xed p oint theory for maps is easier to handle than
the corresp onding p erio dic orbit theory for ows
The construction of a map from a ow via a cross section is gener
ally unique However constructing a ow from a map is generally not
unique Such a construction is called a suspension of the map Studies
of maps and ows are intimately related but they are not identical
For instance a xed p ointofa ow an equilibrium p oint of the di er
ential system has no natural analog in the map setting
A complete accountofPoincar e maps along with a thorough case
study of the Poincar e map for the harmonic oscillator is presented by
Wiggins
Resonance and Hysteresis
Wenow turn our attention to resonance in the Du ng oscillator The
notion of a resonance is a physical concept with no exact mathemat
ical de nition Physically a resonance is a large amplitude resp onse
or output of a system that is sub ject to a xed amplitude input The
concept of a resonance is b est describ ed exp erimentally and resonances
are easy to see in the string apparatus describ ed in section by con
structing a sort of exp erimental bifurcation diagram for forced string
vibrations
Imagine that the string apparatus is running with a small excitation
amplitude the amount of current in the wire is small and a low forcing
frequency the frequency of the alternating current in the wire is much
less than the natural frequency of free wire vibrations To construct
a resonance diagram we need to measure the resp onse of the system
by measuring the maximum amplitude of the string vibrations as a
function of the forcing frequency To do this we slowly increase scan
through the forcing frequency while recording the resp onse of the string
with the optical detectors The results of this exp eriment dep end on
the forcing amplitude as well as where the frequency scan b egins and
ends Decreasing frequency scans can pro duce di erent results from
RESONANCE AND HYSTERESIS
Figure Resp onse curve for a harmonic oscillator
increasing frequency scans
Linear Resonance
Foravery small forcing amplitude the string resp onds with a linear
resonance such as that illustrated in Figure According to linear
theory the resp onse of the string is maximum when
In other words it is maximum when the forcing frequency exactly
equals the natural frequency Aprimary or main resonance exists
when the natural frequency and the excitation frequency are close The
resonance diagram Fig is called a linear resp onse b ecause it can b e
obtained by solving the p erio dically forced linearly damp ed harmonic
oscillator
x x x F cos
which has a general solution of the form
x x e cos
F cos
The constants x and are initial conditions Equation is a
solution to equation if we discard higher order terms in The
maximum amplitude of x as a function of the driving frequency is
found from the asymptotic solution of equation
F cos
lim x
CHAPTER STRING
Figure Schematic of the resp onse curve for a cubic oscillator
which pro duces the linear resp onse diagram shown in Figure since
F
a x max lim x
max
After the transient solution dies out the steady state resp onse has the
same frequency as the forcing term but it is phase shifted by an amount
that dep ends on and F As with all damp ed linear systems the
steady state resp onse is indep endent of the initial conditions so that we
can sp eak of the solution
In the linear solution motions of signi cant amplitude o ccur when
F is large or when Under these circumstances the nonlinear
term in equation cannot b e neglected Thus even for planar
motion a nonlinear mo del of string vibrations may b e required when a
resonance o ccurs or where the excitation amplitude is large
Nonlinear Resonance
A nonlinear resonance curve is pro duced when the frequency is scanned
with a mo derate forcing amplitude F Figure shows the results
of b oth a backward and a forward scan which can b e constructed
from a numerical solution of the Du ng oscillator equation see
App endix C on Ode for a description of the numerical metho ds
The two scans are identical except in the region marked by
l u
Here the forward scan pro duces the upp er branch of the resp onse curve
This upp er branchmakes a sudden jump to the lower branchatthe
frequency Similarly the backward decreasing scan makes a sudden u
RESONANCE AND HYSTERESIS
Figure Nonlinear resonance curveshowing secondary resonances
in addition to the main resonance
jump to the upp er branchat In the region atleast
l l u
two stable p erio dic orbits co exist The sudden jump b etween these two
orbits is indicated bytheupward and downward arrows at and
l u
This phenomenon is known as hysteresis
The nonlinear resp onse curve also reveals several other intriguing
features For instance the maximum resp onse amplitude no longer o c
curs at but is shifted forward to the value This is exp ected
u
in the string b ecause as the string s vibration amplitude increases its
length increases and this increase in length and tension is accompa
nied by a shift in the natural frequency of free oscillations
Several secondary resonances are evident in Figure These sec
ondary resonances are the bumps in the amplitude resonance curvethat
o ccur away from the main resonance
The main resonance and the secondary resonances are asso ciated
with p erio dic orbits in the system The main resonance o ccurs near
when the forcing amplitude is small and corresp onds to the
p erio d one orbits in the system those orbits whose p erio d equals the
forcing p erio d The secondary resonances are lo cated near some ra
tional fraction of the main resonance and are asso ciated with p erio dic
motions whose p erio d is a rational fraction of These p erio dic orbits
denoted by x can often b e approximated to rst order by a sinusoidal
function of the form
m
x A cos
m n
n
where A is the amplitude of the p erio dic orbit m n is its frequency
CHAPTER STRING
Figure Amplitude mo dulated quasip erio dic motions on a torus
and is the phase shift These p erio dic motions are classi ed bythe
integers m and n as follows m n
m an ultraharmonic
n a subharmonic
m n an ultrasubharmonic
Equation is used as the starting p oint for the method of harmonic
balance a pragmatic technique that takes a trigonometric series as the
basis for an approximate solution to the p erio dic orbits of a nonlinear
system see Prob It is also p ossible to have solutions to dif
ferential equations involving frequencies that are not rationally related
Such orbits resemble amplitude mo dulated motions and are generally
known as quasiperiodic motions see Figure A more complete ac
count of nonlinear resonance theory is found in Nayfeh and Mo ok
Parlitz and Lauterb orn also provide several details ab out the nonlinear
resonance structure of the Du ng oscillator
Resp onse Curve
In this section we fo cus on understanding the hysteresis found at the
main resonance of the Du ng oscillator b ecause hysteresis at the main
resonance and at some secondary resonances is easy to observe exp eri
mentally The results in this section can also b e derived by the metho d
of harmonic balance by taking m n in equation how
RESONANCE AND HYSTERESIS
ever we will use a more general metho d that is computationally a little
simpler
We generally exp ect that in a nonlinear system the maximum re
sp onse frequency will b e detuned from its natural frequency An esti
mate for this detuning in the undamp ed free cubic oscillator
x x x
is obtained by studying this equation by the metho d of slow ly varying
amplitude Write
i i
A e A e x
and substitute equation into equation while assuming A
varies slowly in the sense that jA j A Then equation is
approximated by
i A jAj A
where we ignore all terms not at the driving frequency Equation
has a steady state solution in A denoted by a R Inthiscase
jaj
since A and
x a cos
To rst order the strength of the nonlinearity increases the normalized
frequency by an amount dep ending on the amplitude of oscillation and
the nonlinearity parameter This approximate value for a Du ng os
cillator is consistent with the results found in App endix B where exact
solutions for a cubic oscillator are presented
If wewrite A a ib and A a ib then it is easy to
show from Euler s identitythat x a cos b sin Thus the use
of complex numb ers is not essential in this calculation it is merely a trickthat
simpli es some of the manipulations with the sinusoidal functions
CHAPTER STRING
Hysteresis is discovered when we apply the slowly varying amplitude
approximation to the forced damp ed Du ng equation
x x x x F cos
Substituting equation into equation and again keeping
only the terms at the rst harmonic we arrive at the complex amplitude
equation
jAj A F i A i
which in steady state A b ecomes
jAj A F i
To nd the set of real equations for the steady state write the complex
amplitude in the form
i
A ae
where b oth a and are real constants Then equation separates
into two real equations
a F sin
and
a a F cos
which collectively determine b oth the phase and the amplitude of the
resp onse Squaring b oth equations and and then adding
the results we obtain a cubic equation in a
a a F
illustrated in Figure which is known as the response curve In this
approximation the steady state resp onse is given by
x a cos
where a is the maximum amplitude of the harmonic resp onse deter
mined from equation and is the phase shift determined from equations and
RESONANCE AND HYSTERESIS
Hysteresis
The resp onse curveshown in Figure is a plot of a versus cal
culated from equation the nonlinear phase amplitude relation
This curve shows that the string can exhibit hysteresis near a primary
resonance a slowscanofthevariable the so called quasistatic ap
proximation results in a sudden jump b etween the two stable solutions
indicated by the solid lines in Figure The jump from the upp er
branchtothelower branchtakes place at The jump from the lower
u
branch to the upp er branchtakes place at
l
In the parameter regime the resp onse curvereveals
l u
the co existence of three p erio dic orbits at the same frequency but with
di erent amplitudes All these orbits are p ossible solutions to equation
for the values of a indicated in the diagram All three orbits are
harmonic resp onses or p erio d one orbits since their frequency equals
the forcing frequency The middle solution indicated by the dashed
line in Figure is an unstable p erio dic orbit
Basins of Attraction
In a linear system with damping the attracting p erio dic orbit is inde
p endent of the initial conditions In contrast the existence of twoor
more stable p erio dic orbits for the same parameter values in a nonlin
ear system indicates that the initial conditions play a critical role in
determining the system s overall resp onse These attracting p erio dic
orbits are called limit cycles and their global stability is determined by
constructing their basins of attraction A very nice three dimensional
picture of the basins of attraction for the two stable p erio dic orbits
found in the Du ng oscillator is presented by Abraham and Shaw
However this picture of the basins of attraction within the three
dimensional owisvery intricate An equivalent picture of the basins
of attraction constructed with a two dimensional cross section and a
Poincar e map is easier to understand
Aschematic for the basins of attraction in a Du ng oscillator with
three co existing orbits is p ortrayed in Figure The cross section
shows two stable orbits P and P and one unstable orbit P In
the region surrounding the inset of P a small change in the initial
CHAPTER STRING
Figure Schematic of the basins of attraction in the Du ng oscil
lator Adapted from Hayashi
conditions can pro duce a large change in the resp onse of the system
since initial conditions in this region can go to either attracting p erio dic
orbit The unstable p erio dic orbit indicated by P is a saddle xed
pointinthePoincar e map and the stable p erio dic orbits are sinks in
the Poincar e map
The inset of the saddle is the collection of all p oints that approach
P This inset divides the Poincar emapinto two distinct regions the
initial conditions that approach P and the initial conditions that ap
proach P That is the inset to the saddle determines the b oundary
separating the two basins of attraction Again we see the imp ortance
of keeping track of the unstable solutions as well as the stable solu
tions when analyzing a nonlinear system Figure a should b e
compared to but not confused with Figure e the phase plane
for the unforced damp ed Du ng oscillator In the Poincar e map each
xed p oint represents an entire p erio dic orbit not just an equilibrium
pointofthe ow as in Figure More imp ortantly inthePoincar e
atrajectory of the ow map the inset to the sadd le point at P is not
Rather it is the collection of all initial conditions that converge to
P The approach of a single orbit toward P is a sequence of discrete
HOMOCLINIC TANGLES
Figure Schematic of a homo clinic tangle in the Du ng oscillator
p oints indicated by the crosses in Figure In general the inset
and the outset of the saddle represent an in nite continuum of distinct
orbits all of which share a common prop erty namely they arriveat
or depart from a p erio dic p oint of the map
Homo clinic Tangles
Figure shows the Poincar e map for the ow arising in the Du ng
oscillator in a parameter region where hysteresis exists We see that
the inset to P consists of two di erentcurves Similarly the outset
of P also consists of two di erent curves One branch of the outset
approaches P while the other branch approaches P The inset and
the outset of the saddle are not tra jectories in the ow so they can
intersect without violating a fundamental theorem of ordinary di eren
tial equations the unique dep endence of an orbit with resp ect to initial
conditions
It is p ossible to nd parameter values so that the inset and the
outset of the saddle p ointat P do indeed cross A self intersection
of the inset of a saddle with its outset is illustrated in Figure b
and as originally observed byPoincar e it always gives rise to wild
oscillations ab out the saddle see section
Such a self intersection of the inset of a saddle with its outset is
CHAPTER STRING
called a homoclinic intersection and it is a fundamental mechanism by
whichchaos is created in a nonlinear dynamical system The reason is
roughly the following Consider a p ointatacrossingoftheinsetand
the outset indicated by the p oint I in Figure b By de nition
this p oint is part of an orbit that approaches the saddle by b oth its
inset and its outset that is it is doubly asymptotic Consider the
next intersection p ointof I with the cross section P I This p oint
must lie on the inset at a p oint closer to the saddle Next the second
iterate of I under the Poincar e map P I must b e even closer to the
saddle Similarly the preimage of I approaches the saddle along the
outset of the saddle The outset and inset get bunched up near the
saddle creating an image known as a homoclinic tangle Homo clinic
tangles b eat at the heart of chaos b ecause in the region of a homo clinic
tangle initial conditions are sub ject to a violent stretching and folding
pro cess the two essential ingredients for chaos A marvelous pictorial
description of homo clinic tangles along with an explanation as to their
imp ortance in dynamical systems is presented by Abraham and Shaw
Homo clinic tangles are often asso ciated with the existence of strange
sets in a system Indeed it is thought thatinmany instances a strange
attractor is nothing but the closure of the outset of some saddle when
this outset is bunched up in a homo clinic tangle Figure a shows
the cross section for a strange attractor of the Du ng oscillator Figure
b shows the cross section of the outset of the p erio d one saddle in
this strange set for the exact same parameter values The resemblance
between these two structures is striking Indeed developing metho ds to
dissect homo clinic tangles will b e central to the study of chaos in low
problem dimensional nonlinear systems In fact one could call it the
of low dimensional chaos
Nonplanar Motions
Additional dynamical p ossibilities arise when we consider nonplanar
string vibrations These vibrations are also easy to excite with the
string apparatus describ ed in section When a nonmagnetic wire
The closure of a set X is the smallest closed set that is a sup erset of X
NONPLANAR MOTIONS
Figure Comparison of a strange attractor and the outset of a p erio d one saddle in the Du ng oscillator
CHAPTER STRING
Figure Planar p erio dic elliptical nonplanar p erio dic and pre
cessing quasip erio dic motions of a string
is used out of plane motions are observed which are sometimes called
ballo oning or whirling motions These nonplanar vibrations arise even
when the excitation is only planar
Indeed ballo oning motions are hard to avoid Imagine scanning the
forcing frequency of the string apparatus through a resonance The re
sp onse of the string increases as the resonance frequency is approached
and the following b ehavior is typically observed Well b elow the reso
nance frequency the string resp onds with a planar p erio dic oscillation
Fig a As the forcing frequency is increased the amplitude of
the resp onse also grows until the string p ops out of the plane and b e
gins to move in a nonplanar elliptical p erio dic pattern Fig b
That is the string undergo es a bifurcation from a planar to a nonpla
nar oscillation At a still higher frequency the elliptical p erio dic orbit
b ecomes unstable and b egins to precess as illustrated in Figure c
We will present a more complete qualitative account of these whirling
motions in section which is based on the recentwork of Johnson
and Ba ja j Miles and O Reilly Now though we turn our
attention to the whirling motion that o ccurs when no forcing is present
Free Whirli ng
If wepluck a string hard and lo ok closely wetypically see the string
whirling around in an elliptical pattern with a diminishing amplitude
Some understanding of these motions is obtained by considering the free
NONPLANAR MOTIONS
planar oscillations of a string mo deled by the two dimensional equation
r r K r r
which is equation with no forcing term We noted in section
that the linear approximation to equation results in elliptical mo
tion eq We shall use this observation to calculate an approxi
mate solution to equation using a pro cedure put forth byGough
similar results were obtained by Elliot
Transform the problem of nonlinear free vibrations to a reference
frame rotating with an angular frequency with to b e determined
In this rotating frame equation b ecomes
K u u u u u u u
where u is the new radial displacementvector sub jected to the addition
of Coriolis and centrifugal accelerations Let us now lo ok for a solution
of the form
u t x t y t
t
e X cos t X cos t Y sin t Y sin t
where X and Y are small compared to X and Y Lo oking at the
x co ordinate only whenwe substitute equation into equation
and discard appropriate higher order terms we get
X cos t X cos t
t
Y cos t K X cos t Y sin t e X cos t
a similar relation holds for the y co ordinate On equating sinusoidal
terms of the same frequency we after considerable algebra discover
K
t
X Y e
K
t
X Y e
and
t
X Y e Y K X
X Y
CHAPTER STRING
If there is no damping then this approximate solution is
p erio dic in the rotating reference frame and is slightly distorted from
an elliptical orbit The angular frequency is detuned from by
an amount prop ortional to the mean square radius X Y In the
original stationary reference frame equation shows us that the
orbit precesses at a rate prop ortional to the orbital area X Y The
angular frequency in equations to is measured in the
rotating reference frame It is related to the angular frequency in the
stationary reference frame by
K
t
X Y X Y e
Thus in the stationary reference frame the undamp ed motion is
quasip erio dic unless and are accidentally commensurate in which
case the orbit is p erio dic The damp ed oscillations are also elliptical in
character and precess at a rate In b oth cases the detuning given
by equation is due to two sources the nonlinear planar motion
detuning plus a detuning resulting from the precessional frequency
Resp onse Curve
The case of whirling motions sub ject to a planar excitation is describ ed
by the equation
r r K r r A cos t
where the phase space is four dimensional x v y v R The
x y
extended phase space when we add the forcing variable is ve dimen
sional x v y v R S
x y
Equation can b e analyzed for p erio dic motions by a combi
nation of averaging and algebraic techniques not unlike the harmonic
balance metho d Because our text is an exp erimental intro duction to
nonlinear dynamics we present here a qualitative description of the
results of this analysis For further details see references and
As mentioned in the intro duction to this section an exp erimental
frequency scan that passes through a main resonance can result in the
following sequence of motions
NONPLANAR MOTIONS
Figure Resp onse curve for planar and nonplanar motion
Adapted from Johnson and Ba ja j
planar p erio dic nonplanar p erio dic
planar p erio dic
quasip erio dic
precessing elliptical orbit
chaotic
jump to small amplitude planar motions
The basic features of these exp erimental observations agree with those
predicted by equation and are summarized in the resp onse curve
shown in Figure
In the parameter range the resp onse curve indicates
l u
the co existence of three planar p erio dic motions and one nonplanar
p erio dic orbit In this parameter regime the planar p erio dic orbit
b ecomes unstable the string p ops out of the plane and b egins to
execute a whirling motion At some parameter value the nonplanar
q
p erio dic motion itself b ecomes unstable and the system maydoany
numb er of things dep ending on the exact system parameters and ini
tial conditions For instance it may hop to the small amplitude planar
p erio dic orbit Or the ballo oning orbit itself may b ecome unstable and
b egin to precess quasip erio dic motion In addition chaotic motions
can sometimes b e observed in this parameter range These various dy
namical p ossibilities are illustrated schematically in Figure We
CHAPTER STRING
rep eat the motion observed dep ends on the exact system parameters
and the initial conditions b ecause there can b e many co existing attrac
tors with complicated basins of attraction in this region In particular
the chaotic motions are di cult to isolate and observe exp erimentally
without a thorough understanding of the system
Torus Attractor
To construct a cross section for nonplanar p erio dic motion we could
imagine a plot of the p osition of the center of the string and the forcing
phase x y R S Fig A map can b e asso ciated to an
orbit in R S by recording the p osition of the orbit once each forcing
period Although this map is not a true Poincar e map it is easy to
obtain exp erimentally and will b e useful in explaining the notion of a
torus attractor see section
An elliptical p erio dic orbit in the ow is represented in this cross
section by a discrete set of p oints that lie on a closed curve This curve
is top ologically a circle S Fig b Similarly a precessing ellipse
quasip erio dic motion can generate an in nite numb er of p oints these
points ll out this circle Fig c
In the extended space x y this quasip erio dic motion represents
a dense winding of a torus as shown in Figure d Top ologically a
torus is a space constructed from the Cartesian pro duct of two circles
T S S In general an n torus is constructed from n copies of a
circle
n
z
n
S S S T
and a torus attractor naturally arises whenever quasip erio dic motion
is encountered in a dissipative dynamical system The torus is an
A prop er cross section would b e a manifold transverse to the owin R S
i e a four manifold suchas f x v y v j g The torus attractor arises
x y
from a Hopf bifurcation a bifurcation from a xed p ointtoaninvariant curve In
a cross section the limit cycle is represented by a xed p oint At the transition to
quasip erio dic motion this xed p oint loses stabilityandgives birth to an invariant
circle which is a cross section of the torus in the ow
The attracting torus for nonplanar string vibrations is actually a four torus
T
NONPLANAR MOTIONS
Figure Exp erimental cross section for the nonplanar string vibra
tions
attractor b ecause it is an invariant set and an attracting limit set This
is illustrated in Figure whichshows how orbits are attracted to
a torus A graph of a quasip erio dic orbit on a torus attractor is an
amplitude mo dulated time series Fig
Circle Map
We found that the single hump ed map of the interval f I I was
a go o d mo del for some asp ects of the dynamics of the b ouncing ball
system Similarly insight ab out motion near a torus attractor can b e
gained by studying a circle map
g S S
CHAPTER STRING
Figure Torus attractor
where the mapping g most often studied is a two parameter map of the
form
K
sin g
n n n n n
This map has a linear term a constant bias term and a nonlinear
n
term whose strength is determined by the constant K
The frequency of the circle map is monitored bythewinding number
n
g
K
W lim
n
n
If the nonlinear term K equals zero then W The winding number
measures the average increase in the angle p er unit time Fig
An orbit of the circle map is p erio dic if after q iterations
n q
p forintegers p and q The winding numb er for a p erio dic orbit is
n
W p q A quasip erio dic orbit has an irrational winding numb er
Circle maps have a devilish dynamical structure which is explored in reference
NONPLANAR MOTIONS
Figure The winding numb er measures the average increase in the
angle of the circle map
Torus Doubling
A nonlinear system can make a transition from quasip erio dic motion
directly to chaos This is known as the quasiperiodic route to chaos
It is of great practical and historical imp ortance since it was one of
the rst prop osed mechanisms leading to the formation of a strange
attractor
There are in fact many routes to chaos even from a humble T torus
attractor For instance when the T attractor loses stability a stable
higher dimensional torus attractor sometimes forms Another p ossibil
ity in the string system is the formation of a doubled torus illustrated
schematically in Figure In the torus doubling route to chaos our
original torus which is a closed curve in cross section app ears to split
into two circles at the torus doubling bifurcation p oint The torus
doubling route to chaos is reminiscent of the p erio d doubling route to
chaos However it di ers in at least two signi cantways First in most
exp erimental systems there are only a nite numb er of torus doublings
b efore the onset of chaotic motion In fact no more than two torus
doublings haveever b een observed in the string exp eriment Second
the torus doubling route to chaos is a higher dimensional phenomenon
CHAPTER STRING
Figure Schematic of a torus doubling bifurcation
requiring at least a four dimensional ow or a three dimensional map
It is not observed in one dimensional maps unlike the p erio d doubling
route to chaos
Nowthatwehave reviewed some of the more salient dynamical
features of a string s motions let s turn our attention to assembling
the to ols required to view these motions in a real string exp eriment
Exp erimental Techniques
The dynamics of a forced string raises exp erimental challenges common
to a variety of nonlinear systems In this section we describ e a few
of the exp erimental diagnostics that help with the visualization and
identi cation of di erent attractors suchas
equilibrium p oints
limit cycles p erio dic orbits
invariant tori quasip erio dic orbits
strange attractors chaotic orbits
EXPERIMENTAL TECHNIQUES
The main to ols used in the real time identi cation of an attractor are
Fourier p ower sp ectra and real time Poincar e maps In addition the
correlation dimension calculated from an exp erimental time series helps
to con rm the existence of a strange attractor as well as providing a
measure of its fractal structure
The terms strange attractor and chaotic attractor are not al
ways interchangeable Sp eci cally a strange attractor is an attractor
that is a fractal That is the term strange refers to a static geomet
ric prop erty of the attractor The term chaotic attractor refers to an
attractor whose motions exhibit sensitive dep endence on initial condi
tions That is the term chaotic refers to the dynamics on the attractor
We mention this distinction b ecause it is p ossible for an attractor to
b e strange a fractal but not chaotic exhibit sensitive dep endence on
initial conditions Exp erimental metho ds are available for quanti
fying b oth the geometric structure of an attractor fractal dimensions
and the dynamic prop erties of orbits on an attractor Lyapunov exp o
nents
Exp erimental Cross Section
Variables measured directly in the string apparatus include the forcing
phase t and the displacement amplitudes of the string x t and
y t The phase is measured directly from the function generator that
provides the sinusoidal current to the wire The string displacement
is measured from the optical detectors that record the horizontal and
vertical displacement at a xed p oint along the wire In addition the
wire s velo city can b e measured by sending the amplitude displacement
signal through a di erentiator a circuit that takes the analog derivative
of an input signal
Manyapproaches are p ossible for constructing an exp erimental
Poincar e section The particular approachtaken dep ends on b oth the
typ e of system and the equipment at hand Here we assume that the
lab is sto cked with a dual trace storage oscilloscop e and some basic
electronic comp onents A di erent approachmight b e taken for in
stance if wehave access to a digital oscilloscop e either in the form of
a commercial instrument or a plug in b oard to a micro computer
CHAPTER STRING
Figure Schematic for the construction of an exp erimental Poincar e
map for the string apparatus
Planar Cross Section
We will record the Poincar e section on the storage oscilloscop e Our
rst step is to adjust the optical detectors so that the axis for a purely
planar vibration is well aligned with one of the optical detectors Next
the signal from this optical detector is sent to one of the input channels
of the oscilloscop e The other channel is used to record the velo city
via the di erentiator of this same signal Lastly the time base on the
oscilloscop e must b e set to X Y mo de thereby allowing b oth the hori
zontal and vertical oscilloscop e sweeps to b e controlled by the external
signals
The result as shown schematically in the oscilloscop e in Figure
is an exp erimental rendering of the phase space tra jectory for a string
To construct a Poincar e map we need to sample this tra jectory once
each forcing p erio d That is instead of recording the entire tra jectory
weonlywant to record a sequence of p oints on this tra jectory This
can b e accomplished by turning the oscilloscop e s b eam intensity on for
EXPERIMENTAL TECHNIQUES
a brief moment once each forcing p erio d This is easy to do b ecause on
the back of most oscilloscop es is an analog input line lab eled z that
controls the oscilloscop e s b eam intensity Finally we need a triggering
circuit that takes as input the sinusoidal forcing signal and generates as
output a clo ck pulse which is used to brie y turn on the oscilloscop e s
b eam once each forcing p erio d
Triggering Circuit
A simple triggering circuit can b e constructed from a monostable vi
brator and a Schmitt trigger Here though we describ e a slightly
more sophisticated approach based on a phase lo cked lo op PLL The
phase lo cked lo op circuit has the advantage that it allows us to trigger
more than once each forcing p erio d This feature will b e useful when
we come to digitizing a signal b ecause the phase lo cked lo op circuit can
b e used to trigger a digitizer an integer number k times each forcing
period thereby giving us k samples of the tra jectory each cycle
A go o d account of all things electronic including phase lo cked lo ops
is presented in the b o ok by Horowitz and Hill The art of electronics
For our purp oses a phase lo cked lo op chip contains a phase detector
an ampli er and a voltage controlled oscillator VCO in one pack
age A PLL when used in conjunction with a stage counter gener
ates a clo ck signal or triggering pulse that is ideal for constructing a
Poincar e section A schematic of the triggering circuit used with the
string apparatus is presented in Figure and is constructed from
two o the shelf chips a CMOS PLL and a stage counter
This circuit with small adjustments is useful for generating a trigger
ing signal in any forced system The counter is set to one for a Poincar e
section that is it generates one pulse each p erio d It can b e readjusted
to pro duce k pulses p er p erio d when digitizing
Nonplanar Cross Section
To generate an exp erimental cross section for nonplanar motions we
replace the v t input to the oscilloscop e with the output y t fromthe
x
second displacement detector The resulting plot on the oscilloscop e is
prop ortional to the actual horizontal and vertical displacement of the
CHAPTER STRING
Figure Triggering circuit used to construct a Poincar e map in the
string apparatus Courtesy of K Adams and T C A Molteno
string thus providing us with a magni ed view of the string s whirling
Emb edding
In the string system there is no di culty in sp ecifying and measur
ing the ma jor system variables Usually though we are not so lucky
Imagine an exp erimental dynamical system as a blackbox that gen
erates a time series x t In practice wemayknow little ab out the
pro cess inside the blackbox Therefore the exp erimental construction
of a phase space tra jectory oraPoincar e map seems very problematic
One approach to this problem the exp erimental reconstruction of a
phase space tra jectory is as follows We start out by assuming that the
time series is pro duced by a deterministic dynamical system that can
b e mo deled bysomenth order ordinary di erential equation For this
particular example we assume that the system is mo deled by a third
order di erential system as is the case for planar string vibrations
EXPERIMENTAL TECHNIQUES
Then a given tra jectory of the system is uniquely sp eci ed by the value
of the time series and its rst and second derivatives at time t
x t x t x t
This suggests that in reconstructing the phase space we can b egin with
our measured time series and then use x t to calculate two new phase
space variables y x t and z x t de ned by
d
x t y t
dt
d
y t z t
dt
In estimating the p ointwise derivatives of x t in an exp erimentwe
can pro ceed in at least twoways rst we can pro cess the original
signal through a di erentiator and then record digitize the original
signal along with the di erentiated signals or second we could digitize
the signal and then compute the derivatives numerically While b oth
techniques are feasible eachisfraught with exp erimental di culties
b ecause di erentiation is an inherently noisy pro cess This is b ecause
approximating a derivative often involves taking the di erence of two
numb ers that are close in value To see this consider the numerical
derivative of a digitized time series fx t g de ned by
i
x x
i i
y
i
t t
i i
For instance let x x and t t Then
i i i i
y that is the value of the rst derivative is already buried
i
in the noise and the problem just gets worse when taking higher order
derivatives
However let s lo ok at equation again If the sampling time
is evenly spaced then t t is constant so
i i
y x x
i i i
That is almost all the information ab out the derivative is contained in
avariable constructed by taking the di erence of two p oints in the orig
inal time series fx t g This idea can b e generalized as follows Instead i
CHAPTER STRING
of de ning the new variables for the reconstructed phase space in terms
of the derivatives we can recover almost all of the same information
ab out an orbit from the embedded variables de ned by
y x
i i r
z x s r
i i s
where r and s are integers Each new emb edded variable is de ned by
taking a time delay of the original time series Clearly anynumber of
emb edded variables can b e created in this way and this metho d can b e
used to reconstruct a phase space of dimension far greater than three
There are many technical issues asso ciated with the construction of
an emb edded phase space An in depth discussion of these issues can
b e found in reference The rst concern is determining a go o d choice
for the delay times r and s One rule of thumbistotake r to b e small
say or In fact we could de ne the second variable as
v x x
i i i r
and think of it as a velo cityvariable The second emb edding time
should b e much larger than r but not to o large To b e more sp eci c
consider the planar oscillations of a string again In this example a
natural cycle time is given by the p erio d of the forcing term A go o d
choice for s is some sizable fraction of the cycle time For instance let s
say our digitizer is set to sample the signal times each p erio d Then
a sensible choice for r might b e and for s might b e or one quarter
of the forcing p erio d Figure shows a plot of a chaotic tra jectory
in the Du ng oscillator in b oth the original phase space and the phase
space reconstructed from the emb edding variables The similarityof
the two representations lends supp ort to the claim that a tra jectory
in the emb edded phase space provides a faithful representation of the
dynamics
Constructing a real time two dimensional emb edded phase space is
straightforward An emb edded signal is obtained by sending the orig
inal signal through a delay line The current signal and the delayed
signal are sent to the oscilloscop e thereby giving us a real time repre
sentation of the phase space dynamics from our blackbox
EXPERIMENTAL TECHNIQUES
Figure Tra jectory of the Du ng oscillator in a phase space and
b the emb edded phase space with delay time
Power Sp ectrum
A signal from a nonlinear pro cess is a function of time whichwe will
call F t in this section It is p ossible to develop signatures for p erio dic
quasip erio dic and chaotic signals by analyzing the p erio dic prop erties
of F t These signatures which are based on Fourier analysis are
valuable exp erimental aids in identifying di erenttyp es of attractors
Fourier Series
To analyze the p erio dic prop erties of F t it is useful to uncover a func
tional representation of the signal in terms of the orthogonal functions
cos t L and sin t L of p erio d L To determine the Fourier se
ries for F t wemust nd the constants a and b so that the following
k k
identity holds
a
a cos t b sin t F t
L L
a cos t b sin t
L L
Fourier showed that the constants a and b in equation are
k k
computed from F t by means of the integral formulas
Z
L
F t cos kt dt k a
k
L L
CHAPTER STRING
Figure Digitized time series
and
Z
L
kt dt k F t sin b
k
L L
Finite Fourier Series
In equations we are assuming that F t is a continuous
function of t Equations and dictate that to calculate the
k th constants a and b in the Fourier series eq we substitute
k k
F t into the previous equations and integrate In exp eriments with
digitized data the signal we actually work with is an equally spaced
discrete set of p oints F t measured at the set of times ft g Fig
p p
Therefore we need to develop a discrete analog to the Fourier
series the nite Fourier series
Let us consider an even number of points p er p erio d N Thenthe
N sample p oints are measured at
L L N L
N N N
or more succinctly
pL
t p N
p
N
The nite Fourier series for a function sampled at F t is
p
N
X
A A
N
A cos F t kt B sin kt cos Nt
k k
L L L
k
EXPERIMENTAL TECHNIQUES
where
N
X
A F t cos kt k N
k p p
N L
p
and
N
X
F t sin kt k N B
p p k
N L
p
In the ab ove formulas we assumed that the function is sampled
at N p oints that the value at the p oint N is L and that the
endp oints and L satisfy the p erio dic b oundary condition
F F L
If the latter assumption do es not hold the convention is to average the
values at the endp oints so that the value at F is taken to b e
F F L
In this case the formulas for the co e cients are
N
X
F L F
F t cos A kt
p k p
N L
p
and
N
X
B kt F t sin
k p p
N L
p
Equations and can b e viewed as a transform or mapping
That is given a set of numb ers F t these relations generate twonew
p
sets of numb ers A k and B k It is easy to program this transform
The co de to calculate the discrete Fourier transform of F t involves
p
a double lo op see eqs and the inner lo op cycles through
the index k and the outer lo op covers the index p Each lo op has
order N steps so the total numb er of computations is of order N
The discrete Fourier transform is therefore quite slow for large N
say N see App endix D for programs to calculate discrete
Fourier transforms Fortunately there exists an alternativemethod
CHAPTER STRING
for calculating the Fourier transform called the fast Fourier transform
or FFT The FFT is an N ln N computation whichismuch faster than
the discrete Fourier for large data sets A detailed explanation of the
FFT along with C co de examples is found in Numerical Recipes
Power Sp ectrum
The amplitude co e cients A and B give us a measure of howwell the
k k
signal ts the k th sinusoidal term A plot of A versus k or B versus
k k
k is called a frequency spectrum The normalized p ower sp ectrum
amplitude is de ned by
q
A B H
k
k k
AplotofH versus k is called the power spectrum This graphical
k
representation tells us howmuch of a given frequency is in the original
signal
Exp erimental p ower sp ectra can b e obtained in at least three ways
i obtain a sp ectrum analyzer or signal analyzer which is a commercial
instrument dedicated to displaying real time p ower sp ectra ii obtain
a signal analyzer card for a micro computer this is usually less exp ensive
than option i or iii digitize your data and write an FFT for your
computer this option is the cheap est Having successfully pro cured
sp ectra capabilities wenowmove on to describing how to use them
Sp ectral Signatures
The sp ectral signatures for p erio dic quasip erio dic and chaotic motion
are illustrated in Figure The p ower sp ectrum of a p erio d one orbit
is dominated by one central p eak call it The p ower sp ectrum of a
period two orbit also has a sharp p eak at and additional p eaks at the
subharmonic and the ultrasubharmonic These new sp ectral
All things are fair in love war and exp erimental physics Metho ds to obtain
a sp ectrum analyzer may include a lo cating and nationalizing a sp ectrum
analyzer from a nearby lab oratory or b lo cating and appropriating a sp ectrum
analyzer on grounds of national security Another option is to use an older mo del
sp ectrum analyzer suchasaTektronix L sp ectrum analyzer which plugs into an
older mo del Tek scop e
EXPERIMENTAL TECHNIQUES
Figure Time series and p ower sp ectra a p erio dic p erio d one
b p erio dic p erio d two c quasip erio dic d chaotic
CHAPTER STRING
p eaks are the sidebands ab out the primary frequency that come into
existence through say a p erio d doubling bifurcation of the p erio d one
orbit More generally thepower sp ectrum of a p erio d n orbit consists
of a collection of discrete p eaks showing the primary frequency and its
overtones In p erio dic motion all the p eaks are rationally related to
the primary p eak resonance
Quasip erio dic motion is characterized by the co existence of two in
commensurate frequencies Thus the p ower sp ectrum for a quasip eri
o dic motion is made up from at least two primary p eaks and
which are not rationally related Additionally each of the primary
p eaks can have a complicated overtone sp ectrum The mixing of the
overtone sp ectra from and usually allows one to distinguish p e
rio dic motion from quasip erio dic motion A quick examination of the
time series or the Poincar e section can also help to distinguish p erio dic
motion from quasip erio dic motion
The p ower sp ectrum of a chaotic motion is easy to distinguish from
p erio dic or quasip erio dic motion Chaotic motion has a broad band
power sp ectrum with a rich sp ectral structure The broad band nature
of the chaotic p ower sp ectrum indicates the existence of a continuum
of frequencies A purely random or noisy pro cess also has a broad band
power sp ectrum so weneedtodevelop metho ds to distinguish noise
from chaos In addition to its broad band feature a chaotic p ower
sp ectrum can also havemany broad p eaks at the nonlinear resonances
of the system These nonlinear resonances are directly related to the
unstable p erio dic orbits emb edded within the chaotic attractor So the
power sp ectrum of a chaotic attractor do es provide some limited infor
mation concerning the dynamics of the system namely the existence of
unstable p erio dic orbits nonlinear resonances that strongly in uence
the recurrence prop erties of the chaotic orbit
Additionally in the p erio dic and quasip erio dic regimes new humps
and p eaks can app ear in a p ower sp ectrum whenever the system is
near a bifurcation p oint These sp ectral features are called transient
precursors of a bifurcation A detailed theory of these precursors with
many practical applications has b een develop ed by Wiesenfeld and co
workers
EXPERIMENTAL TECHNIQUES
Attractor Identi cation
So far wehave discussed four measurements that allow us to visualize
and identify the attractor coming from a nonlinear pro cess
time series
power sp ectra
phase space p ortrait or reconstructed phase space
exp erimental Poincar e sections
All these qualitative techniques can b e set up with instruments that
are commonly available in any lab oratory
To get a time series we ho ok up the output signal from the nonlinear
pro cess to an input channel of the oscilloscop e and use the time base
of the scop e to generate the temp oral dimension of the plot To obtain
apower sp ectrum we use a sp ectrum analyzer or digitize the data and
use an FFT An exp erimental phase space p ortrait can b e plotted on
an oscilloscop e either by recording two system variables directly such
as x y or x x or from the delayed variable x t x t where
is the delay time Lastly in a forced system the Poincar e section
is obtained from the phase space tra jectory by strobing it once each
forcing p erio d using the z blanking on the back of the oscilloscop e
Nowtoidentify an attractor we monitor these four diagnostic to ols
as wevary a system parameter A bifurcation p ointiseasytoiden
tify by using these to ols and the existence of a particular bifurcation
sequence say a sequence of p erio d doubling bifurcations is a strong
indicator for the p ossible existence of chaotic motion
The use of these diagnostic to ols is illustrated schematically in Fig
ure for the p erio d doubling route to chaos For the parameter
values and an examination of any one of these diagnostics
is su cient to identify the existence of a p erio dic motion as well as its
period For these four diagnostics as well as the fact that the
c
strange Poincar e section arose from a sequence of bifurcations from a
p erio dic state supp ort the claim that the motion is chaotic
In particular the Poincar e section is useful for distinguishing low
dimensional chaos from noise The chaotic Poincar e section illustrated
CHAPTER STRING
Figure Perio d doubling route to chaos a p erio d one b p erio d
two c p erio d four and d chaotic Adapted from Tredicce and Abraham
EXPERIMENTAL TECHNIQUES
Figure Poincar e maps from p erio dic quasip erio dic chaotic and
noisy random pro cesses
in Figure resembles the familiar one hump ed map studied in Chap
ter In contrast the Poincar e map for a noisy signal from a sto chastic
pro cess lls the whole oscilloscop e screen with a random collection of
dots see Fig Thus motion on a strange attractor has a strong
spatial correlation not present in a purely random signal In the next
section we will quantify this observation byintro ducing the correlation
dimension a measure that allows us to distinguish whether the signal
is coming from a low dimensional strange attractor or from noise
The quasip erio dic route to chaos is illustrated in Figure In
this case the phase p ortrait for a quasip erio dic motion would resemble
a Lissa jous pattern whichmay b e hard to distinguish from a slowly
evolving chaotic orbit The Poincar e map on the other hand is useful
in distinguishing b etween these two cases The sequence of dots forming
the Poincar e map in the quasip erio dic regime lie on a closed curvethat
is easy to distinguish from the spread of p oints in the Poincar e map for
a strange attractor
Correlation Dimension
One di erence b etween a chaotic signal from a strange attractor and a
signal from a noisy random pro cess is that p oints on the chaotic attrac
tor are spatially organized One measure of this spatial organization is
the correlation integral
number of pairs i j whose distance jy y j C lim
i j
n n
CHAPTER STRING
Figure Quasip erio dic route to chaos a equilibrium b p eri
o dic c quasip erio dic and d chaotic Adapted from Tredicce and Abraham
EXPERIMENTAL TECHNIQUES
where n is the total numb er of p oints in the time series This correlation
function can b e written more formally by making use of the Heaviside
function H z
n
X
H jy y j C lim
i j
n
n
i j
where H z for p ositive z and otherwise Typically the vector
y used in the correlation integral is a p ointintheemb edded phase
i
space constructed from a single time series according to
y x x x x i
i i i r i r
i m r
For a limited range of it is found that
C
that is the correlation integral is prop ortional to some p ower of
This p ower is called the correlation dimension and is a simple mea
sure of the p ossibly fractal size of the attractor
The correlation integral gives us an e ective pro cedure for assigning
a fractal dimension to a strange set This fractal dimension is a simple
way to distinguish a random signal from a signal generated by a strange
p ossibly chaotic set In principle a random pro cess has an in nite
correlation dimension Intuitively this is b ecause an orbit of a random
pro cess is not exp ected to haveany spatial structure In contrast the
correlation dimension for a closed curve a p erio dic orbit is and for
atwo dimensional surface such as quasip erio dic motion on a torus
is A strange fractal set can have a correlation dimension that is
not an integer For instance the strange set arising at the end of the
p erio d doubling cascade found in the quadratic map has a correlation
dimension of indicating that the dimension of this strange
attractor is somewhere b etween that of a nite collection of p oints
and a curve
There exist some technical issues asso ciated with the calculation of
a correlation dimension from a time series that have to do with the
choice of the emb edding dimension m and the delaytimer These
issues are dealt with more fully in references and There now
String
exist several computer co des in the public domain that have to a large
extent automated the calculation of from a single time series x t
i
Such a time series could come from a simulation or from an exp eriment
See for example the BINGO co de by Albano or the e cient algo
rithm discussed by Grassb erger Thus the correlation dimension
is now a standard to ol in a nonlinear dynamicist s to olb ox that helps
one distinguish b etween noise and low dimensional chaos
References and Notes
J R Tredicce and N B Abraham Exp erimental measurements to identify
and or characterize chaotic signals in Lasers and quantum optics edited by
L M Narducci E J Quel and J R Tredicce CIF Series Vol World
Scienti c New Jersey N Gershenfeld An exp erimentalist s intro duc
tion to the observation of dynamical systems in Directions in Chaos Vol
edited by Hao B L World Scienti c New Jersey pp N B
Abraham A M Albano and N B Tu llaro Intro duction to measures of
complexity and chaos in NATO ASI Series B Physics Pro ceedings of the
international workshop on Quantitative Measures of Dynamical Complexityin
Nonlinear Systems Bryn Mawr PA USA June edited byN B
Abraham A M Albano T Passamante and P Rapp Plenum Press New
York J Crutch eld D Farmer N Packard R Shaw G Jones and R
Donnelly Power sp ectral analysis of a dynamical system Phys Lett A
C Gough The nonlinear free vibration of a damp ed elastic string J Acoust
So c Am
The original version of this apparatus was constructed and examined for p ossi
ble chaotic motions by T C A Molteno at the University of Otago Dunedin
New Zealand For more details see T C A Molteno and N B Tu llaro
Torus doubling and chaotic string vibrations Exp erimental results J Sound
Vib
R C Cross A simple measurement of string motion Am J Phys
D Martin Decay rates of piano tones J Acoust So c Am
R Hanson Opto electronic detection of string vibrations The Physics Teacher
March
References and Notes
N B Tu llaro Nonlinear and chaotic string vibrations Am J Phys
N B Tu llaro Torsional parametric oscillations in wires Eur
J Phys
For theoretical and exp erimental details on chaos in an elastic b eam see F C
Mo on Chaotic vibrations An introduction for applied scientists and engineers
John Wiley Sons New York In particular App endix C contains
a detailed description of the construction of an inexp ensive b eam exp eriment
showing chaos Some discussion of the Du ng equation as it relates to the
motion of a b eam can also b e found in J M Thompson and H B Steward
Nonlinear dynamics and chaos John Wiley Sons New York
J Elliot Intrinsic nonlinear e ects in vibrating strings Am J Phys
J Elliot Nonlinear resonance in vibrating strings Am J
Phys
See Chapter of A Nayfeh and D Mo ok Nonlinear oscil lators Wiley
Interscience New York A more elementary account of p erturbative
solutions to the Du ng equation is presented byA Nayfeh Introduction to
perturbation techniques John Wiley Sons New York
A classic in nonlinear studies is C Hayashi s Nonlinear oscil lations in physical
systems Princeton University Press Princeton NJ The metho d of
harmonic balance is used extensively byHayashi see pages
and for a discussion and examples of this technique applied to the
Du ng oscillator
A real gem of mathematical exp osition is V I Arnold s Ordinary di erential
equations MIT Press Cambridge MA The basic theorems of ordinary
di erential equations are proved in Chapter Arnold s text is the b est intro
ductory exp osition to the geometric study of di erential equations available A
must read for all students of dynamical systems
S Wiggins Introduction to applied nonlinear dynamical systems and chaos
Springer Verlag New York pp
N B Tu llaro and G A Ross Ode A program for the numerical solution of
ordinary di erential equations Bell Lab oratories Technical Memorandum
program and do cumentation are in the public domain Also
see App endix C
U Parlitz and W Lauterb orn Sup erstructure in the bifurcation set of the
Du ng equation Phys Lett A U Parlitz and W
Lauterb orn Resonance and torsion numb ers of driven dissipative nonlinear
oscillators Z Naturforsch a
String
The metho d of slowly varying amplitude is also known as the averaging metho d
of Krylov Bogoliub ov Mitrop olsky see E A Jackson Perspectives in nonlin
ear dynamics Vol Cambridge University Press New York pp
In the optics community it is also known as the rotating wave ap
proximation see P W Milonni M L Shih and J R Ackerhalt Chaos in
laser matter interactions World Scienti c Singap ore pp
This is a bit sneaky A A here b ecause there is no damping term causing
a frequency shift in the approximation to rst order In general A A and
then the slowly varying amplitude approximation results in b oth a detuning
and a phase shift Both data are enco ded by A C
Amarvelous pictorial intro duction to dynamical systems theory is the unique
series of b o oks by R Abraham and C Shaw Dynamics The geometry of
behavior Vol Aerial Press Santa Cruz CA Volume pp
contains a go o d description of the geometry of the phase space asso ciated
with hysteresis in the Du ng oscillator and Volume pp shows how
homo clinic tangles are formed
J M Johnson and A K Ba ja j Amplitude mo dulated and chaotic dynamics
in resonant motion of strings J Sound Vib
J Miles Resonant nonplanar motion of a stretched string J Acoust So c
Am
O M O Reilly Thechaotic vibration of a string Ph D thesis Cornell Univer
sity
P Bak The devil s staircase Physics Today P Bak T
Bohr and M H Jensen Mo de lo cking and the transition to chaos in dissipative
systems Physica Scripta Vol T P Cvitanovi c B Shraiman
and B S oderb erg Scaling laws for mode lockings in circle maps Physica Scripta
For a fuller account of torus attractors and quasip erio dic motion in a nonlinear
system see Chapter of P Berg e Y Pomeau and C Vidal Order within
chaos John Wiley Sons New York An elementary discussion of
fractal dimensions and the correlation integral is found on pages
A Arn edo P Coullet and E Spiegel Cascade of p erio d doublings of tori Phys
Lett A K Kaneko Doubling of torus Prog Theor Phys
F Argoul and A Arn edo From quasip erio dicity
to chaos an unstable scenario via p erio d doubling bifurcation or tori J de
M ecanique Th eorique et Appliqu ee Numer ero sp ecial
Problems
W Ditto M Spano H Savage S Rauseo J Heagy and E Ott Exp erimental
observation of a strange nonchaotic attractor Phys Rev Lett
N Packard J Crutch eld J Farmer and R Shaw Geometry from a time
series Phys Rev Lett Also see Chapter of D Ruelle
Chaotic evolution and strange attractors Cambridge University Press New
York For a geometric approach to the emb edding problem see Th
Buzug T Reimers and G P ster Optimal reconstruction of strange attractors
from purely geometrical arguments Europhys Lett
R Hamming An introduction to applied numerical analysis McGraw Hill
New York
W Press B Flannery S Teukolsky and W Vetterling Numerical recipes in
C Cambridge University Press New York
K Wiesenfeld Virtual Hopf phenomenon A new precursor of p erio d doubling
bifurcations Phys Rev A K Wiesenfeld Noisy precursors
of nonlinear instabilities J Stat Phys P BryantandK
Wiesenfeld Suppression of p erio d doubling and nonlinear parametric e ects
in p erio dically p erturb ed systems Phys Rev A B
McNamara and K Wiesenfeld Theory of sto chastic resonance Phys Rev A
K Wiesenfeld and N B Tu llaro Suppression of p erio d
doubling in the dynamics of the b ouncing ball Physica D
P Grassb erger and I Pro caccia Characterization of strange attractors Phys
Rev Lett P Grassb erger and I Pro caccia Measuring the
strangeness of strange attractors Physica D
The BINGO program runs on the IBM PC For a copycontact A Albano
DepartmentofPhysics Bryn Mawr College Bryn Mawr PA
P Grassb erger An optimized b ox assisted algorithm for fractal dimensions
Phys Lett A This brief pap er provides a Fortran program
for calculating the correlation integral For another new approach to calculating
the correlation integral see Xin Jun Hou Rob ert Gilmore Gabriel B Mindlin
and Hern an Solari An e cient algorithm for fast O N ln N b ox counting
Phys Lett
Problems Problems for section
String
Use Amp ere s law to nd the magnetic eld at a radial distance r from a long
straight current carrying wire
The Joule heating lawsays that the p ower dissipated by a current carrying
ob ject is
P VI I R
where V is the voltage I is the current and R is the resistance Furthermore
for small temp erature changes Temp the fractional change of length of a
solid ob eys
L
Temp
L
where is the linear co e cient of thermal expansion of the material Discuss
the relevance of these twophysical laws on the string apparatus
Section
From equation show that the lo cation of the two equilibrium p oints is
q
l l
approximately given by r l
l
Solve the di erential equationx x x and graph x t versus t for a
few values of Whyis called the damping co e cient
Solve the di erential equation r r r x y Draw a plot
x y
of r t x t y t for a xed
x y
Verify that the variables in equation are dimensionless
Verify that equation is a solution to equation with f t
and K discard second order terms in Draw the solution r t in the
x y plane
Derive equation from Figure Hint To account for the factor of
realize that each spring makes a separate contribution to the restoring force
Section
a Show that the equation of motion for a simple p endulum in dimensionless
variables is
sin
b Write this as a rst order system with v S R Using equation
nd the p otential energy function and a few integral curves for a p endulum
Problems
c Show that the p endulum has equilibrium p oints at and and
discuss the stability of these xed p oints by relating them to the con gura
tions of the physical p endulum Are there any saddle p oints Are there any
centers
d Drawa schematic of the phase plane Identify the separatrix in this phase
p ortrait Orbits inside the separatrix are called oscil lations Why Orbits
outside the separatrix are called rotations Why
e Add a dissipative term to get a damp ed p endulum and discuss how
this changes the phase p ortrait In particular discuss the relation of the
insets and outsets of the equilibrium p oints with the basins of attraction Are
there any attractors Are there any rep ellers
f Now add a forcing term f cos t and write the di erential equations for
the system in the extended phase space v R S S where t
De ne a global cross section for the forced damp ed p endulum see eq
Section
Verify that equation is a solution to equation discard higher
order terms in
Plot the linear resp onse curve a eq for a few representativevalues
of F and
This exercise illustrates the metho d of harmonic balance Assume an approx
imate solution of the form
x X sin Y cos
to the Du ng equation
a Substitute x into equation and equate terms containing sin and
cos separately to zero
b Showthat
AX Y X AY F
where
R R X Y A
c Show that
A R F
d Plot the resp onse R versus for and several values
of F Plot the amplitude characteristic R versus F for
and Indicate the unstable solution with a dashed line Hint It is
easier to plot R as the indep endentvariable
String
Section
Verify equations for the free whirling motions of a string
Write a program to iterate the circle map eq and explore its solutions
for di erentvalues of K and
Section
For a time series fx g in whichthe x are sampled at evenly spaced times
i i
write a transformation b etween the phase space variable y x x t
i i i i
t and the emb edded phase space variable y x withr Whyis
i i i r
the emb edded phase space rotated like it is in Figure Whydoesone
usually cho ose r
Write a program based on equations to calculate the discrete
Fourier amplitude co e cients and the p ower sp ectrum eq for a dis
crete time series Test the program on some sample functions for p erio dic
motion e g x t cos t cos t and quasip erio dic motions e g
p
t x t cos t cos
Chapter
Dynamical Systems Theory
Intro duction
This chapter is an eclectic mix of standard results from the mathemat
ical theory of dynamical systems along with practical results terminol
ogy and notation useful in the analysis of a low dimensional dynamical
system The examples in this chapter are usually con ned to two
dimensional maps and three dimensional ows
In the rst three chapters we presented nonlinear theory byway
of examples Wenow present the theory in a more general setting
Many of the fundamental ideas have already b een illustrated in the
one dimensional setting For example in one dimension we found that
the stability of a xed p oint is determined by the derivative at the
xed p oint The same result holds in higher dimensions However the
actual computational machinery needed is far more intricate b ecause
the derivativeofan n dimensional map is an n n matrix
Another key idea wehave already intro duced is hyp erb olicity hyp er
b olic sets and symb olic analysis see section In this chapter we
present a detailed study of the Smale horsesho e which is the canonical
example of a hyp erb olic chaotic invariant set A thorough under
standing of this example is essential to the analysis of a chaotic rep eller
or attractor We conclude this chapter with a discussion of sensitive
dep endence on initial conditions This brings us back to the Lyapunov
exp onent whichwe de ne for a general n dimensional dynamical sys
CHAPTER DYNAMICAL SYSTEMS THEORY tem
INTRODUCTION
Maps Flows
Lorenz Equation
Quadratic Map
x y z R R
f x R R
x y x
x x x
n n n
y x y xz
parameter
z z xy
parameters
Sine Circle Map
f S S
Du ng Equation
K
sin
n n n
x v R S R S
parameters
x v
K
v x x v f cos
Henon Map
nonautonomous form
f x y R R
x x x x f cos t
x x y
n n
parameters
n
y x
n n
f
parameters
Forced Damp ed Pendulum
v S S R S S R
Baker s Map
v
f x y I I I I
v v sin f cos
x x mo d
n n
y for x
n n
parameters
y
n
y for x
n n
f
parameter
Mo dulated Laser
u z R S R S
IHJM Optical Map
u z f cos u
f z C C
z z z u
i h
z Bz exp i
n n
jz j
n
parameters
parameters
f
B
CHAPTER DYNAMICAL SYSTEMS THEORY
Flows and Maps
Most of the dynamical systems studied in this b o ok are either three
dimensional ows or one or two dimensional maps Common examples
of maps and ows are listed on the previous page
Flows are sp eci ed by di erential equations section Simi
larly maps are sp eci ed by di erence equations
x f x
n n
Maps can also b e written as x f x The notation is read
n
as maps to The forward orbit of x is O x ff x n g
n
where f f f is the nth comp osite of f and f is the identity
function If the inverse f is well de ned then the backward orbit of
n
x is O x ff x n g Finally the orbit of x is the sequence
S
of all p ositions visited by x O x O x O x
Flows
A rst order system of di erential equations is written as
dx
f x t
dt
where x x x x are the dep endentvariables t is the inde
n
p endentvariable time and is the set of parameters for the system
Sometimes the parameter dep endence is denoted by a subscript as in
f x t
n n
A vector eld is formally de ned bya mapF A R R
that assigns a vector F x toeach p oint x in its domain A More
generally a vector eld on a manifold M is given by a map that assigns
avector to eachpointinM For most of this chapter we only need to
n
work with R A system governed by a time indep endentvector eld
is called autonomous otherwise it is called nonautonomous Aswesaw
in section any nonautonomous vector eld can b e converted to
an autonomous vector eld of a higher dimension
The ow of the vector eld F is analytically de ned by
x t F x t
t
x x
FLOWS AND MAPS
Figure a The ow of a solution curve b The ow of a collec
tion of solution curves resulting in a continuous transformation of the
manifold
The p osition x is the initial condition or initial state The initial con
dition is also written as x when it is sp eci ed at t A solution
curve trajectory or integral curve of the ow is an individual solution
of the ab ove di erential equation based at x We often explicitly note
the time dep endence of the p osition the solution curve to the ab ove
di erential equation by writing x t when wewant to indicate the p o
sition of the tra jectory at a time t The collection of all states of a
dynamical system is called the phase space
The term ow describing the evolution of the system in phase
space comes by analogy from the motion of a real uid ow The
ow x t is regarded as a function of the initial condition x and the
single parameter time t The ow tells us the p osition of the initial
condition x after a time t As illustrated in Figure a the p osition
of the p ointonthe ow line through x is carried or ows to the p oint
x t A geometric description of a owsays that it is a one parameter
family of di eomorphisms of a manifold That is the ow lines of the
ow which are solution curves of the di erential equation provide a
continuous transformation of the manifold into itself Fig b
The ow is often written as x to highlight its dep endence on
t
the single parameter t The ow is also known as the evolution
t
operator Comp osition of the evolution op erator is de ned in a natural
way starting at the state x at time s it rst ows to the p oint
CHAPTER DYNAMICAL SYSTEMS THEORY
x s and then onto the p oint x s t The evolution
s t s
op erator satis es the group prop erties
i identity ii
t s t s
which are taken as the de ning relations of a ow for an abstract dy
namical system If the system is nonreversible we sp eak of a semi ow
A semi ow ows forward in time but not backward
An explicit example of an evolution op erator is given by solving the
linear di erential equationx x Here the vector eld is found from
the equivalent rst order system x v v x so that the vector
eld is
F x v v x
This vector eld generates a ow that is a simple rotation ab out the
origin see Fig The evolution op erator is given by the rotation
matrix
cos t sin t x x t
x
t
sin t cos t v v t
Poincar eMap
A continuous ow can generate a discrete map in at least twoways by
a time T map and byaPoincar emap Atime T map results when a
ow is sampled at a xed time interval T That is the ow is sampled
whenever t nT for n andsoon
The more imp ortantway as describ ed for instance in Gucken
heimer and Holmes in whicha continuous ow generates a discrete
n
map is via a Poincar e map Let b e an orbit of a ow in R As
t
illustrated in Figure it is often p ossible to nd a local cross sec
n
tion R ab out which is of dimension n The cross section
need not b e planar however it must b e transverse to the ow All the
orbits in the neighborhood of must pass through The technical
requirement is that F x N x for all x where N x isthe
x v
The comp onents of the map should b e written as where the sup er
script indicates the dep endent co ordinate It is a common convention though to
suppress the and mix the dep endentvariables with the co ordinate functions so
x v
that t t x t v t
FLOWS AND MAPS
Figure Construction of a Poincar e map from a lo cal cross section
unit normal vector to at x Let p be a pointwhere intersects
and let q beapoint in the neighb orho o d of p Then the Poincar e
map or rst return map isde nedby
P P q q
where q is the time taken for an orbit starting at q to return
to It is useful to de ne a Poincar e map in the neighb orho o d of a
p erio dic orbit If the orbit is p erio dic of p erio d T then p T
A p erio dic orbit that returns directly to itself is a xed p oint of the
Poincar e map Moreover an orbit starting at q close to p will havea
return time close to T
For forced systems such as the Du ng oscillator studied in sec
tion a global cross section and Poincar e map are easy to de ne
since the phase space top ology is R S All p erio dic orbits of a
forced system have a p erio d that is an integer multiple of the forcing
period In this situation it is sensible to pick a planar global cross
section that is transverse to S see Fig The return time for this
cross section is indep endent of p osition and equals the forcing p erio d
In this sp ecial case the Poincar e map is equivalent to a time T map
The Poincar e map for the example of the rotational ow generated
by F x v v x is particularly trivial A go o d cross section is
de ned by the p ositive half of the x axis f x v jv andx g
All the orbits of the ow are xed p oints in the cross section so the
Poincar e map is just the identitymap P x x
CHAPTER DYNAMICAL SYSTEMS THEORY
Figure a Construction of a susp ension of a map b The sus
p ension is not unique an arbitrary number of twists can b e added
See App endix E H enon s Trick for a discussion of the numerical
calculation of a Poincar e map from a cross section
Susp ension of a Map
A discrete map can also b e used to generate a continuous ow A
canonical construction for this is the so called suspension of a map
which is in a sense the inverse of a Poincar e map Given a discrete map
f of an n dimensional manifold M itisalways p ossible to construct a
owonann dimensional manifold formed by the Cartesian pro duct
R with the original manifold M R M This susp ension
pro cess is illustrated in Figure a where weshow a mapping and
the ow formed by susp ending this map Each sequence of p oints
of the map b ecomes an orbit of the ow with the prop erty that if
f x x then f x x The original map is recovered
n t n T t
from the susp ended owby a time T map The susp ension construction
The susp ension is de ned globally while the cross section for a Poincar e
map is only de ned lo cally Thus these two constructions are not completely
complementary
ASYMPTOTIC BEHAVIOR AND RECURRENCE
is far from unique For instance as illustrated in Figure b we
could add an arbitrary numb er of complete twists to this particular
susp ension and still get an identical time T map The numb er of full
twists in the susp ended ow is called the global torsion and it is a
top ological invariant of the ow indep endent of the underlying map
Creed and Quest
The close connection b etween maps and ows Poincar emapsand
susp ensions gives rise to The Discrete Creed
Anything that happ ens in a ow also happ ens in a lower
dimensional discrete dynamical system and conversely
The creed is stated in the mo de of the sunshine patriot leaving plenty
of ro om to duck as necessary This creed is implicitinPoincar e s
original work but was rst clearly enunciated by Smale
The essence of dynamical systems studies is stated in The Dynam
ical Quest
Where do orbits go and what do they do see when they
get there
The dynamical quest emphasizes the top ological i e qualitative char
acterization of the long term b ehavior of a dynamical system
Asymptotic Behavior and Recurrence
In this section we present some more mathematical vo cabulary that
helps to re ne our notions of invariant sets limit sets attractors and
rep ellers asymptotic b ehavior and recurrence For the most part we
state the fundamental de nitions in terms of maps The corresp ond
ing de nitions for ows are completely analogous and can b e found in
Wiggins
Recurrence is a key theme in the study of dynamical systems The
simplest notion of recurrence is p erio dicity Recall that a p ointofamap
n
is periodic of period n if there exists an integer n such that f p p
i
and f p p i n This notion of recurrence is to o restricted
CHAPTER DYNAMICAL SYSTEMS THEORY
since it fails to account for quasip erio dic motions or strange attractors
We will therefore explore more general notions of recurrence Atthe
end we will argue that the chain recurrent set is the b est de nition
of recurrence that captures most of the interesting dynamics
Invariant Sets
Formally a set S is an invariant set of a ow if for any x S wehave
x t S for all t R S is an invariant set of a map if for any
n
x S f x S for all n We also sp eak of a positively invariant set
when we restrict the de nition to p ositive times t orn
Invariant sets are imp ortant b ecause they give us a means of de
comp osing phase space If we can nd a collection of invariant sets
then we can restrict our attention to the dynamics on eachinvariant
set and then try to sew together a global solution from the invariant
pieces Invariant sets also act as b oundaries in phase space restricting
tra jectories to a subset of phase space
Limit Sets and Nonwandering
We b egin byintro ducing the limit set which starts us down the road
toward de ning an attractor Let p be a pointofamapf M M of
the manifold M Then the limit set of p is
n
i
p fa M j there exists a sequence n suchthat f p ag
i
Conversely by going backwards in time we get the limit set of p
n
i
p r g p fr M j there exists a sequence n such that f
i
These limit sets are the closure of the ends of the orbits For ex
ample if p is a p erio dic p oint then p O p It is not di cult to
show that p is a closed subset of M and that it is invariant under
f i e f p p Finally a point p M is called recurrent if it
is part of the limit set i e p p
The forward limit set L is de ned as the union of all limit sets
likewise the backward limit set L is de ned as the union of all limit
ASYMPTOTIC BEHAVIOR AND RECURRENCE
sets
p p and L f L f
p M p M
The forward and backward limit sets are not necessarily closed
This observation motivates yet another useful notion of recurrence the
nonwandering set A p oint p M wanders if there exist a neighb orho o d
n
U of p and an m suchthatf U U for all n m A
nonwandering point isonethatdoesnotwander This brings us to the
nonwandering set which is a closed invariant underf subset of
M
f fq M j q is nonwandering g
The dynamical decomp osition of a set into its wandering and non
wandering parts separates in mathematical terms a dynamical sys
tem into its transientbehavior the wandering set and long term or
asymptotic b ehavior the nonwandering set
The limit set and the nonwandering set do not address the ques
tion of the stability of an asymptotic motion To get to the idea of an
attractor we b egin with the idea of an attracting set A closed invari
ant set A M is an attracting set if there is some neighb orho o d U of
A such that for all x U and all n
n n
f x U and f x A
Moreover the domain or basin of attraction of A is given by
n
f U
n
The attracting set can consist of a collection of di erent sets that are
dynamically disconnected For instance a single attracting set could
consist of two separate p erio dic orbits Toovercome this last di culty
we will say that an attractor is an attracting set that contains a dense
orbit Conversely a repel ler is de ned as a rep elling set with a dense
orbit
Although this is a reasonable de nition mathematically we will see
that this is not the most useful de nition of an attractor for physical
applications or numerical simulations In these circumstances it will
turn out that a more useful alb eit mathematically naive de nition of
CHAPTER DYNAMICAL SYSTEMS THEORY
an attractor or rep eller is the closure of a certain collection of p erio dic
orbits The prop er de nition of an attractor is yet another hot sp ot in
the creative tension b etween the rigor demanded by a mathematician
and the utility required bya physicist
Chain Recurrence
In the previous section we attempted to capture all the recurrentbe
havior of the mapping f M M Let
Per f fp erio dic p oints of f g
S
p L f
p M
f fnonwandering p oints g
Clearly
Per f L f f
There is no universally accepted notion of a set that contains all the
recurrence but this set ideally oughttobeclosedandinvariant Per f
is to o small p erio dicity is to o limited a kind of recurrence L f
is not necessarily closed f while closed and invariant has the
drawback that
f j
The mapping f j makes sense of course b ecause of s
invariance
In certain contexts the chain recurrent set whichisbiggerthan
has all the desirable prop erties According to the chain recurrent p oint
of view all the interesting dynamics take place in the chain recurrent
set
Let An pseudo orbit is a nite sequence suchthat
d f x x i n
i i
where d is a metric on the manifold M An pseudo orbit can
b e thought of as a computer generated orbit b ecause of the slight
roundo error the computer makes at each stage of an iteration see Fig a
ASYMPTOTIC BEHAVIOR AND RECURRENCE
Figure a An pseudo orbit or computer orbit of a map b a
chain recurrent p oint
A p oint x M is chain recurrent if for all there exists an
pseudo orbit x x x such that x x x see Fig b
n n
The chain recurrent set is de ned as
R f fchain recurrent p ointsg
The chain recurrent set R f is closed and f invariant Moreover
R For pro ofs see reference or
As an example consider the quadratic map f x x x for
p
and M It is easy to see from graphical
analysis that if x then f g is the attracting limit set When
x the limit set is a Cantor set see section and the
dynamics on are top ologically conjugate to a full shift on two symbols
so the p erio dic orbits are dense in The chain recurrentset R f
S
f g
As another example consider the circle map f S S shown
in Figure in which the only xed p oints are x and y Thearrows
indicate the direction a p oint go es in when it is iterated It is easy to
see that
fx y g Per f L L
CHAPTER DYNAMICAL SYSTEMS THEORY
Figure Circle map with two xed p oints
However R f S b ecause an pseudo orbit can jump across the
xed p oints Chain recurrence is a very weak form of recurrence
Expansions and Contractions
In this section we consider howtwo dimensional maps and three
dimensional ows transform areas and volumes Do es a map lo cally
expand or contract a region Do es a ow lo cally expand or contract a
volume in phase space Toanswer each of these questions we need to
calculate the Jacobian of the map and the divergence of the ow
In this section we are concerned with showing how to do these calcula
tions In section where weintro duce the tangent map weprovide
some geometric insightinto these calculations
DerivativeofaMap
n m
To x notation let f b e a map from R to R sp eci ed by m functions
f f f
m
n m
of n variables Recall that the derivative of a map f R R at x
is written as T Df x and consists of an m n matrix called the
matrix of partial derivatives of f at x
f f
x x
n
Df x
f f
m m
x x
n
Some b o oks call this the di erential of f and denote it by df x
EXPANSIONS AND CONTRACTIONS
Figure Deformation of an in nitesimal region under a map
The derivativeof f at x represents the best linear approximation to f
near to x
As an example of calculating a derivative consider the function from
R to R that transforms p olar co ordinates into Cartesian co ordinates
f r f r f r r cos r sin
The derivative of this particular transformation is
f f
cos r sin
r
Df r
f f
sin r cos
r
Jacobian of a Map
The derivative contains essential information ab out the lo cal dynamics
of a map In Figure weshowhow a small rectangular region R
of the plane is transformed to f R under one iteration of the map
f u v R R where f u v x u v and f u v y u v
The Jacobian of f written x y u v is the determinant of the
derivative matrix Df x y off
x x
y x y x y x
u v
y y
u v u v v u
u v
See Marsden and Tromba for the n dimensional de nition
CHAPTER DYNAMICAL SYSTEMS THEORY
In the example just considered x y r cos r sin the Jacobian
is
x y
r cos sin r
r
The Jacobian of a map at x determines whether the area ab out x
expandsorcontracts If the absolute value of the Jacobian is less than
one then the map is contracting if the absolute value of the Jacobian
is greater than one then the map is expanding
A simple example of a contracting map is provided bytheH enon
map for the parameter range In this case
y f x y x x
n n n n
n
f x y y x
n n n n
and a quick calculation shows
x y
n n
x y
n n
The Jacobian is constant for the H enon map it do es not dep end on
the initial p osition x y When iterating the H enon map the area is
multipliedeach time by and after k iterations the size of an initial
area a is
k
a a j j
In particular if then the area is contracting
Divergence of a Vector Field
Recall from a basic course in vector calculus that the divergence of a
vector eld represents the lo cal rate of expansion or contraction p er
unit volume So to nd the lo cal expansion or contraction of a ow
wemust calculate the divergence of a vector eld The divergence of a
three dimensional vector eld F x y z F F F is
F F F
div F r F
x y z
Let V b e the measure of an in nitesimal volume centered at x
Figure shows howthisvolume evolves under the ow the diver
EXPANSIONS AND CONTRACTIONS
Figure Evolution of an in nitesimal volume along a owline
gence of the vector eld measures the rate at which this initial volume
changes
d
V t j div F x
t
V dt
For instance the divergence of the vector eld for the Lorenz system
is
y x F
x y xz r F r
z xy F
we nd that the ow is globally contracting at a constant rate whenever
the sum of and is p ositive
Dissipative and Conservative
The lo cal rate of expansion or contraction of a dynamical system can b e
calculated directly from the vector eld or di erence equation without
explicitly nding any solutions Wesay a system is conservative if the
absolute value of the Jacobian of its map exactly equals one or if the
divergence of its vector eld equals zero
r F
CHAPTER DYNAMICAL SYSTEMS THEORY
for all times and all p oints Aphysical system is dissipative if it is
not conservative Most of the physical examples studied in this b o ok
are dissipative dynamical systems The phase space of a dissipative
dynamical system is continually shrinking onto a smaller region of phase
space called the attracting set
Equation of First Variation
Another quantity that can b e calculated directly from the vector eld
is the equation of rst variation which provides an approximation for
the evolution of a region ab out an initial condition x The ow x t
is a function of b oth the initial condition and time We will often b e
concerned with the stability of an initial p oint in phase space and thus
we are led to consider the variation ab out a p oint x while holding the
time t xed Let D denote di erentiation with resp ect to the phase
x
variables while holding t xed Then from the di erential equation for
a ow eq we nd
x t D F x t D
x x
t
which on applying the chain rule on the right hand side yields the
equation of rst variation
D x t DF x t D x t
x x
t
This is a linear di erential equation for the op erator D DF x t
x
is the derivativeof F at x t If the vector eld F is n dimensional
then b oth D F and D are n n matrices
x x
Turning once again to the vector eld
F x v v x
we nd that the equation of rst variation for this system is
x x x x
x v x v
v v
v v
x v
x v
Note that this de nition of dissipative can include expansive systems These
will not arise in the physical examples considered in this b o ok See Problem
FIXED POINTS
i
The sup erscript i to indicates the ith comp onentof ow and the
subscript j to denotes that we are taking the derivative with resp ect
j
v v
The comp o to the j th phase variable For example
x
x
nents of are the co ordinate p ositions so they could b e rewritten as
x v
t t x t v t The dot as always denotes di erentiation
with resp ect to time
Fixed Points
An equilibrium solution of a vector eld x f x isapoint x that do es
not change with time
f x
Equilibria are also known as xed p oints stationary p oints or steady
state solutions A xedpoint of a map is an orbit which returns to itself
after one iteration
x f x x
We will tend to use the terminology xed p oint when referring to
a map and equilibrium when referring to a ow The theory for
equilibria and xed p oints is very similar Keep in mind though that
a xed p oint of a map could come from a p erio dic orbit of a ow This
section brie y outlines the theory for ows The corresp onding theory
for maps is completely analogous and can b e found for instance in
Rasband
Stability
At least three notions of stability apply to a xed p oint lo cal stability
global stability and linear stability Here we will discuss lo cal stability
and linear stability Linear stability often but not always implies lo cal
stability The additional ingredient needed is hyp erb olicity This turns
out to b e quite general hyp erb olicity plus a linearization pro cedure is
usually su cient to analyze the stability of an attracting set whether
it b e a xed p oint p erio dic orbit or strange attractor
The notion of the lo cal stability of an orbit is straightforward A
xed p oint is lo cally stable if solutions based near x remain close to
CHAPTER DYNAMICAL SYSTEMS THEORY
Figure a A stable xed p oint b An asymptotically stable xed
point
x for all future times Further if the solution actually approaches
the xed p oint i e x t x as t then the orbit is called
asymptotical ly stable Figure a shows a center that is stable but
not asymptotically stable Centers commonly o ccur in conservative
systems Figure b shows a sink an asymptotically stable xed
point that commonly o ccurs in a dissipative system A xed p ointis
unstable if it is not stable A saddle and source are examples of unstable
xed p oints see Fig
Linearization
To calculate the stability of a xed p oint consider a small p erturbation
y about x
x x t y
The Taylor expansion substituting eq into eq ab out x
gives
x x t y f x t Df x t y higher order terms
It seems reasonable that the motion near the xed p oint should b e
governed bythe linear system
y Df x t y
FIXED POINTS
since x t f x t If x t x is an equilibrium p oint then Df x
is a matrix with constantentries We can immediately write down the
solution to this linear system as
y t exp Df x t y
where exp Df x is the evolution op erator for a linear system If welet
A Df x denote the constant n n matrix then the linear evolution
op erator takes the form
A t A t exp At id At
where id denotes the n n identity matrix
The asymptotic stability of a xed p oint can b e determined by the
eigenvalues of the linearized vector eld Df at x In particular wehave
the following test for asymptotic stability an equilibrium solution of a
nonlinear vector eld is asymptotically stable if all the eigenvalues of
the linearized vector eld Df x have negative real parts
If the real part of at least one eigenvalue exactly equals zero and
all the others are strictly less than zero then the system is still linearly
stable but the original nonlinear system mayormay not b e stable
Hyp erb olic Fixed Points Saddles
Sources and Sinks
Let x x b e an equilibrium p ointofa vector eld Then x is called
a hyperbolic xed p ointifnone of the real parts of the eigenvalues of
Df x is equal to zero The test for asymptotic stability of the previous
section can b e restated as a hyp erb olic xed p oint is stable if the real
parts of all its eigenvalues are negative A xedpointofamapis
hyp erb olic if none of the mo duli of the eigenvalues equals one
The motion near a hyp erb olic xed p oint can b e analyzed and
broughtinto a standard form by a linear transformation to the eigen
vectors of Df x Additional analysis including higher order terms is
usually needed to analyze the motion near a nonhyp erb olic xed p oint
At last we can precisely de ne the terms saddle sink source and
center A hyp erb olic equilibrium solution is a sadd le if the real part of
CHAPTER DYNAMICAL SYSTEMS THEORY
Figure Complex eigenvalues for a two dimensional map with a
hyp erb olic xed p oint a saddle b sink and c source
at least one eigenvalue of the linearized vector eld is less than zero and
if the real part of at least one eigenvalue is greater than zero Similarly
a sadd le point for a map is a hyp erb olic p oint if at least one of the
eigenvalues of the asso ciated linear map has a mo dulus greater than
one and if one of the eigenvalues has mo dulus less than one
Ahyp erb olic p ointofa owisastable node or sink if all the eigen
values have real parts less than zero Similarly if all the mo duli are
less than one then the hyp erb olic p ointofamapisasink
Ahyp erb olic p ointisanunstable node or source if the real parts of
all the eigenvalues are greater than zero The mo duli of a source of a
map are all greater than one
A center is a nonhyp erb olic xed p oint for which all the eigenvalues
are purely imaginary and nonzero mo dulus one for maps For a pic
ture of the elementary equilibrium p oints in three dimensional space see
Figure of Thompson and Stewart The corresp onding stability
information for a hyp erb olic xed p ointofatwo dimensional map is
summarized in Figure
Invariant Manifolds
According to our discussion of invariant sets in section wewould
like to analyze a dynamical system by breaking it into its dynamically
invariant parts This is particularly easy to accomplish with linear
INVARIANT MANIFOLDS
systems b ecause we can write down a general solution for the ow
tA
op erator as e see section The eigenspaces of a linear owor
map i e the spaces formed by the eigenvectors of A are invariant
subspaces of the dynamical system Moreover the dynamics on each
subspace are determined by the eigenvalues of that subspace If the
n
original manifold is R theneachinvariant subspace is also a Euclidean
n
manifold which is a subset of R It is sensible to classify each of these
invariant submanifolds according to the real parts of its eigenvalues
i
s
E is the subspace spanned by the eigenvectors of A with Re
i
c
E is the subspace spanned by the eigenvectors of A with Re
i
u
E is the subspace spanned by the eigenvectors of A with Re
i
s c
E is called the stable space of dimension n E is called the center
s
u
space of dimension n and E is called the unstable space of dimension
c
n If the original linear manifold is of dimension n thenthesumof
u
the dimensions of the invariant subspaces must equal n n n n
u c s
n This de nition also works for maps when the conditions on are
i
s c
replaced by modulus less than one E modulus equal to one E and
u
modulus greater than one E
For example consider the matrix
B C
A
A
This matrix has eigenvalues and eigenvectors
and the ow on the invariant manifolds is illustrated
in Figure
Center Manifold Theorem
It is imp ortanttokeep in mind that wealways sp eak of invariantman
ifolds based at a p oint This p oint is usually a xed p oint x ofa ow
or a p erio dic p oint of a map In the linear setting the invariantman
ifold is just a linear vector space In the nonlinear setting we can also
de ne invariant manifolds that are not linear subspaces but are still
CHAPTER DYNAMICAL SYSTEMS THEORY
Figure Invariant manifolds for a linear ow with eigenvalues
i
s c u
E E E
n
manifolds That is lo cally they lo ok like a copyofR These invariant
manifolds are a direct generalization of the invariant subspaces of the
linear problem They are the most imp ortant geometric structure used
in the analysis of a nonlinear dynamical system
The way to generalize the notion of an invariant manifold from the
linear to the nonlinear setting is straightforward In b oth the linear and
nonlinear settings the stable manifold is the collection of all orbits that
approachapoint x Similarly the unstable manifold is the collection of
all orbits that depart from x The fact that this notion of an invariant
manifold for a nonlinear system is well de ned is guaranteed bythe
center manifold theorem
Center Manifold Theorem for Flows Let f x bea
n
smo oth vector eld on R with f x and A Df x
The sp ectrum set of eigenvalues f g of A divides into
i
three sets and where
s c u
Re
s i
Re
i c i
Re
u i
s c u
Let E E andE b e the generalized eigenspaces of
s c
and There exist smo oth stable and unstable manifolds
u
s u s u
called W and W tangentto E and E at x and a center
INVARIANT MANIFOLDS
Figure Invariant manifolds of a saddle for a two dimensional map
c c s c
manifold W tangentto E at x The manifolds W W
u
and W are invariant for the ow The stable manifold
s u
W and the unstable manifold W are unique The center
c
manifold W need not b e unique
s c
W is called the stable manifold W is called the center manifold and
u
W is called the unstable manifold A corresp onding theorem for maps
also holds and can b e found in reference or Numerical metho ds
for the construction of the unstable and stable manifolds are describ ed
in reference
Always keep in mind that ows and maps di er a tra jectory of a
n
owisacurvein R while the orbit of a map is a discrete sequence of
p oints The invariant manifolds of a ow are comp osed from a union of
solution curves the invariant manifolds of a map consist of a union of a
discrete collection of p oints Fig The distinction is crucial when
we come to analyze the global b ehavior of a dynamical system Once
again we reiterate that the unstable and stable invariant manifolds
are not a single solution but rather a collection of solutions sharing a
common asymptotic past or future
An example from Guckenheimer and Holmes where the invari
ant manifolds can b e explicitly calculated is the planar vector eld
x x y y x
This system has a hyp erb olic xed p oint at the origin where the lin
CHAPTER DYNAMICAL SYSTEMS THEORY
Figure a Invariant manifolds at the origin for the linear ap
proximation b Invariant manifolds for the original nonlinear system
earized vector eld is
x x y y
The stable manifold of the linearized system is just the y axis and the
unstable manifold of the linearized system is the x axis Fig a
Returning to the nonlinear system we can solvethissystemby elimi
nating time
dy y x c y
x or y x
x dx x x
where c is a constantofintegration It is now easy to see Prob
that Fig b
x
u s
W f x y j y g and W f x y j x g
Homo clinic and Hetero clinic Points
We informally de ne the unstable manifold and the stable manifold for
ahyp erb olic xed p oint x of a map f by
s n
W x fx j lim f x x g
n
INVARIANT MANIFOLDS
Figure a Poincar e map in the vicinity of a p erio dic orbit
s u
W x W x b The map shown with a transversal intersection
at a homo clinic p oint
and
u n
W x fx j lim f x x g
n
These manifolds are tangent to the eigenvectors of f atx
We are led to study a two dimensional map f by considering the
Poincar e map of a three dimensional ow in the vicinity of a p erio dic
orbit This situation is illustrated in Figure The map f is orienta
tion preserving b ecause it comes from a smo oth ow The p erio dic or
bit of the owgives rise to the xed p oint x of the map The xed p oint
s
x has a one dimensional stable manifold W x and a one dimensional
u
unstable manifold W x Poincar ewas led to his discovery of chaotic
behavior and homo clinic tangles see section and App endix H by
considering the interaction b etween the stable and unstable manifold
ofx One p ossible interaction is shown in Figure a where the un
s u
stable manifold exactly matches the stable manifold W x W x
However such a smo oth match is exceptional The more common
p ossibilityisfora transversal intersection between the stable and unsta
Informally a map of a surface is orientation preserving if the normal vector to the surface is not ipp ed under the map
CHAPTER DYNAMICAL SYSTEMS THEORY
Figure a A homo clinic p oint b A hetero clinic p oint
ble manifold Fig b The lo cation of the transversal intersection
is called a homoclinic point when b oth the unstable and stable manifold
emanate from the same p erio dic orbit Fig a The intersection
point is called a heteroclinic point when the manifolds emanate from dif
ferent p erio dic orbits A hetero clinic p ointisshown in Figure b
where the unstable manifold emanating fromx intersects the stable
manifold of a di erent xedpoint y
The existence of a single homo clinic or hetero clinic p oint
forces the existence of an in nityofsuch p oints Moreover
it also gives rise to a homo clinic hetero clinic tangle This
tangle is the geometric source of chaotic motions
To see why this is so consider Figure A homo clinic p oint
is indicated at x This homo clinic p oint is part of b oth the stable
manifold and the unstable manifold
s u
x W x and x W x
Also shown is a p oint a that lies on the stable manifold b ehind x the
direction is determined by the arrow on the manifold i e a x on
s
W Similarly the p oint b lies on the unstable manifold with b x on
u
W Now wemust try to nd the lo cation of the next iterate of f x
sub ject to the following conditions
The map f is orientation preserving
s u
f x W x and f x W x all the iterates of a homo
clinic p oint are also homo clinic p oints
INVARIANT MANIFOLDS
s
Figure The interaction of the stable manifold W x and the
u
unstable manifold W x with a homo clinic p oint x The homo clinic
p oint x gets mapp ed to the homo clinic p oint f x The orientation
of the map is determined by considering where p oints a and b in the
vicinityof x are mapp ed
s u
f a f x on W and f b f x onW
A picture consistent with these assumptions is shown in Figure
The p oint f x must lie at a new homo clinic p oint that is at a new
intersection p oint ahead of x The rst candidate for the lo cation
of f x is the next intersection p oint indicated at d However f x
could not b e lo cated here b ecause that would imply that the map f
is orientation reversing see Prob The next p ossible lo cation
which do es satisfy all the ab ove conditions is indicated by f x More
complicated constructions could b e envisioned that are consistent with
the ab ove conditions but the solution shown in Figure is the sim
plest
Now f x is itself a homo clinic p oint And the same argument
applies again the p oint f x must lie closer tox and ahead of f x
Fig a In this way a single homo clinic orbit must generate
an in nite numb er of homo clinic orbits This sequence of homo clinic
p oints asymptotically approachesx
Since f arises from a ow it is a di eomorphism and thus invertible
CHAPTER DYNAMICAL SYSTEMS THEORY
Figure a Images and preimages of a homo clinic p oint b A
homo clinic tangle resulting from a single homo clinic p oint
Therefore exactly the same argument applies to the preimages ofx
n
That is f x approachesx via the unstable manifold The end
s u
result of this construction is the violent oscillation of W and W in
the region ofx These oscillations form the homo clinic tangle indicated
schematically in Figure b
The situation is even more complicated than it initially app ears
The homo clinic p oints are not p erio dic orbits but Birkho and Smith
showed that each homo clinic p oint is an accumulation p oint for an in
nite family of p erio dic orbits Thus each homo clinic tangle has an
in nite numb er of homo clinic p oints and in the vicinityofeach homo
clinic p oint there exists an in nite numb er of p erio dic p oints Clearly
one ma jor goal of dynamical systems theory and nonlinear dynamics
is the developmentoftechniques to dissect and classify these homo
clinic tangles In section wewillshowhow the orbit structure of a
homo clinic tangle is organized by using a horsesho e map In Chapter
we will continue this top ological approachbyshowing how knot theory
can b e used to unravel a homo clinic tangle
EXAMPLE LASER EQUATIONS
Example Laser Equations
Wenow consider a detailed example to help reinforce the barrage of
mathematical de nitions and concepts in the previous sections Our
example is taken from nonlinear optics and is known as the laser rate
equation
F u zu
F
F z z z u
In this mo del u is the laser intensityand z is the p opulation inversion
The parameters and are damping constants When certain lasers
are turned on they tend to settle down to a constantintensitylightout
put constant u after a series of damp ed oscillations ringing around
the stable steady state solution This b ehavior is predicted by the laser
rate equation
To calculate the stability of the steady states we need to know the
derivativeof F
F F
z u
u z
A A
DF
F F
z u
u z
Steady States
The steady states are found by setting F
zu
z z u
These equations havetwo equilibrium solutions The rst whichwe
lab el a o ccurs at
a u z
The second whichwe lab el b o ccurs at
z u b
CHAPTER DYNAMICAL SYSTEMS THEORY
The lo cation in the phase plane of these equilibrium p oints is shown in
Figure
The motion in the vicinityofeach equilibrium p oint is analyzed by
nding the eigenvalues and eigenvectors v
v A v
of the derivative matrix of F A DF at each xed p ointofthe ow
At the p oint a we nd
A
DFj
a
Eigenvalues of a Matrix
To calculate the eigenvalues of DF we recall that the general solution
for the eigenvalues of any real matrix
a a
A
a a
are given by
p p
tr A tr A
where
tr A a a
det A a a a a
A tr A det A
Applying these formulas to DFj we nd
a
tr DFj
a
det DFj
a
DFj
a
EXAMPLE LASER EQUATIONS
and the eigenvalues for the xed p oint a are
and
The eigenvalue is p ositive and indicates an unstable direction the
eigenvalue is negative and indicates a stable direction The xed
p oint a is a hyp erb olic saddle
Eigenvectors
s
The stable and unstable directions and hence the stable space E a
u
and the unstable space E a are determined by the eigenvectors of
DFj The stable direction is calculated from
a
A
Solving this system of simultaneous equations for and gives the
unnormalized eigenvector for the unstable space as
v m
m
Similarly the stable space is found from the eigenvalue equation for
A
Solving this set of simultaneous equations shows that the stable space
is just the z axis
v
In fact the z axis is invariant for the whole ow i e if z thenu
s s
for all t sothez axis is the global stable manifold at a E a W a
CHAPTER DYNAMICAL SYSTEMS THEORY
Figure Phase p ortrait for the laser rate equation
Stable Focus
To analyze the dynamics in the vicinity of the xed p oint b we need to
nd the eigenvalues and eigenvectors of
DFj
b
The eigenvalues of DFj are
b
i and i
where
p
and
The xed p oint b is a stable fo cus since the real parts of the eigenvalues
are negative This fo cus represents the constantintensity output of a
laser and the oscillation ab out this steady state is the ringing a laser
initially exp eriences when it is turned on
The global phase p ortrait pieced together from lo cal information
ab out the xed p oints is pictured in Figure
SMALE HORSESHOE
Smale Horsesho e
In section we stressed the imp ortance of analyzing the orbit struc
ture arising within a homo clinic tangle From a top ological and physical
p oint of view analyzing the orbit structure primarily means answering
two questions
What are the relative lo cations of the p erio dic orbits
How are the stable and unstable manifolds interwoven within a
homo clinic tangle
By studying the horsesho e example we will see that these questions are
intimately connected
In sections and weanswered the rst question for the
one dimensional quadratic map by using symb olic dynamics For the
sp ecial case of a chaotic hyp erb olic invariant set to b e discussed in
section Smale found an answer to b oth of the ab ove questions for
maps of any dimension Again the solution involves the use of symb olic
dynamics
The prototypical example of a chaotic hyp erb olic invariant set is the
Smale horsesho e A detailed knowledge of this example is essential
for understanding chaos The Smale horsesho e like the quadratic map
for is an example of a chaotic rep eller It is not an attractor
Physical applications prop erly fo cus on attractors since these are di
rectly observable It is therefore sometimes b elieved that the chaotic
horsesho e has little use in physical applications In Chapter we will
show that such a b elief could not b e further from the truth Remnants
of a horsesho e sometimes called the proto horsesho e are buried
within a chaotic attractor The horsesho e or some other variantof
ahyp erb olic invariant set acts as the skeleton on whichchaotic and
p erio dic orbits are organized To quote Holmes Horsesho es in a sense
provide the backb one for the attractors Therefore horsesho es
are essential to b oth the mathematical and physical analysis of a chaotic system
CHAPTER DYNAMICAL SYSTEMS THEORY
From Tangles to Horsesho es
The horsesho e map is motivated by studying the dynamics of a map in
the vicinity of a p erio dic orbit with a homo clinic p oint Such a system
gives rise to a homo clinic tangle Consider a small b ox AB C D inthe
vicinity of a p erio dic orbit as seen from the surface of section This
situation is illustrated in Figure a The b oxischosen so that the
side AD is part of the unstable manifold and the sides AB and DC
are part of the stable manifold Wenow ask how this b oxevolves under
forward and backward iterations The unstable and stable manifolds
of the p erio dic orbit are invariant Therefore when the b ox is iterated
anypoint of the b ox that lies on an invariant manifold must always
remain on this invariant manifold
If we iterate p oints in the b oxforward then we generally end up af
ter a nite numb er of iterations with the horsesho e shap e C D A B
shown in Figure b The initial segment AD which lies on the un
stable manifold gets mapp ed to the segment A D which is also part
of the unstable manifold Similarly ifwe iterate the b oxbackward
we nda backward horsesho e p erp endicular to the forward horsesho e
Fig c The b ox of initial p oints gets compressed along the un
u s
stable manifold W and stretched along the stable manifold W After
a nite numb er of iterations the forward image of the b oxwillinter
sect the backward image of the b ox Further iteration pro duces more
intersections
Each new region of intersection contains a p erio dic orbit see Fig
as well as segments of the unstable and stable manifolds That is
the horsesho e can b e viewed as generating the homo clinic tangle Smale
realized that this typ e of horsesho e structure o ccurs quite generally in
achaotic system Therefore he decided to isolate this horsesho e map
from the rest of the problem
Aschematic for this isolated horsesho e map is presented in Figure
d Like the quadratic map it consists of a stretch and a fold
The horsesho e map can b e thought of as a thickened quadratic map
Unlike the quadratic map though the horsesho e map is invertible The
future and past of all p oints are well de ned However the itinerary
of p oints that get mapp ed out of the b ox are ignored We are only
concerned with p oints that remain in the b ox under all future and past
SMALE HORSESHOE
Figure Formation of a horsesho e inside a homo clinic tangle
iterations These p oints form the invariantset
Horsesho e Map
A mathematical discussion of the horsesho e map is provided byDe
vaney or Wiggins Here we will present a more descriptive
account of the horsesho e that closely follows Wiggins s discussion
The forward iteration of the horsesho e map is shown in Figure
CHAPTER DYNAMICAL SYSTEMS THEORY
Figure a Forward iteration of the horsesho e map b Backward iteration of the horsesho e map
SMALE HORSESHOE
a The horsesho e is a mapping of the unit square D
f D R D f x y R j x y g
whichcontracts the horizontal directions expands in the vertical direc
tion and then folds The mapping is only de ned on the unit square
Points that leave the square are ignored Let the horizontal strip H
b e all p oints on the unit square with y and let horizontal
strip H b e all p oints with y Then a linear horseshoe
map is de ned by the transformation
x x
f H
y y
x x
f H
y y
where and The horsesho e map takes the horizontal
strip H to the vertical strip V f x y j x g and H to the
vertical strip V f x y j x g
f H V and f H V
The strip H is also rotated by The inverse of the horsesho e map
f is shown in Figure b The inverse map takes the vertical
rectangles V and V to the horizontal rectangles H and H
The invariant set of the horsesho e map is the collection of all
p oints that remain in D under all iterations of f
n
f D f D f D D f D f D
n
This invariant set consists of a certain in nite intersection of horizontal
and vertical rectangles Tokeep track of the iterates of the horse
sho e map the rectangles we will need the symbols s f g with
i
i The symb olic enco ding of the rectangles works much
the same way as the symb olic enco ding of the quadratic map see sec
tion
The rst forward iteration of the horsesho e map pro duces twover
tical rectangles called V and V V is the vertical rectangle on the
CHAPTER DYNAMICAL SYSTEMS THEORY
Figure a Forward iteration of the horsesho e map and symb olic
names b Backward iteration c Symb olic enco ding of the invariant
points constructed from the forward and backward iterations
SMALE HORSESHOE
left and V is the vertical rectangle on the right The next step is to
apply the horsesho e map again thereby pro ducing f D As shown
in Figure a V and V pro duce four vertical rectangles lab eled
from left to right V V V and V Applying the map yet again
pro duces eightvertical strips lab eled V V V V V V
n
V andV In general the nth iteration pro duces rectangles
The lab eling for the vertical strips is recursively de ned as follows if
the current strip is left of the center then a is added to the frontof
the previous lab el of the rectangle if it falls to the right a is added
So for instance the rectangle lab eled V starts on the right The rect
angle lab eled V originates from strip V but it currently lies on the
left Lastly the strip V starts on the right then go es to the left
and then returns to the right again To eachvertical strip we asso ciate
a symbolic itinerary
V
s s s s s
n
i
whichgives the approximate orbit left or right of a vertical strip after
n iterations The minus sign in the symb olic lab el indicates that the
symbol s arises from considering the ith preimage of the particular
i
vertical strip under f Also note that the vertical strips get progres
n
sively thinner so that after n iterations each strip has a width of
n
The backward iterates pro duce horizontal strips at the nth it
n
eration The heightofeach of these horizontal strips is From
the two horizontal strips H and H the inverse map f pro duces
four horizontal rectangles lab eled from b ottom to top as H H
H andH Fig b This in turn pro duces eight horizontal
rectangles H H H H H H H andH Each
horizontal strip can b e uniquely lab eled with a sequence of s and s
H
s s s s s
i n
where the symbol s indicates the currentapproximate lo cation b ot
tom or top of the horizontal rectangle The fact that the lab eling
scheme is unique follows from the de nition of f and the observation
that all of the horizontal rectangles are disjoint Unlikethevertical
strips the indexing for the horizontal strips starts at and is p ositive
The need for this indexing convention will b ecome apparentwhenwe
sp ecify the lab eling of p oints in the invariantset
CHAPTER DYNAMICAL SYSTEMS THEORY
Now the invariant set of the horsesho e map f is given by the in
nite intersection of all the horizontal and vertical strips The invariant
set is a fractal in fact it is a pro duct of twoCantor sets The map f
generates a Cantor set in the horizontal direction and the inverse map
f generates a Cantor set in the vertical direction The invariantset
is in a sense the pro duct of these twoCantor sets
Symb olic Dynamics
Wecanidentify p oints in the invariant set according to the following
scheme After one forward iteration and one backward iteration the in
variant set is lo cated within the four shaded rectangular regions shown
in Figure c After twoforward iterations and twobackward it
erations the invariant set is a subset of the shaded regions The
shaded regions are the intersection of the horizontal and vertical strips
To each shaded region we asso ciate a bi in nite symb ol sequence
s s s s s s s s
n n
constructed from the lab el of the vertical and horizontal strips forming
a p ointintheinvariant set The right hand side of the symb olic name
s s s s is the lab el from the horizontal strip H The
n s s s s
n
left hand side of the symb olic name s s s s is the lab el
n
from the vertical strip written backwards V For in
s s s s s
n
i
stance the shaded region lab eled L in Figure c has a symb olic
name The to the right of the dot indicates horizontal
strip H The backwards to the left of the dot indicates
that the shaded region comes from the vertical strip V
We hone in closer and closer to the invariantsetby iterating the
horsesho e map b oth forward and backward Moreover the ab ove la
b eling scheme generates a symb olic name or symb olic co ordinate for
eachpoint of the invariant set This symb olic name contains informa
tion ab out the dynamics of the invariant p oint
To see how this works more formally let us call the symb ol space
of all bi in nite sequences of s and s A metric on b etween the
two sequences
s s s s s s
n n
SMALE HORSESHOE
s s s s s s
n n
is de ned by
X
if s s
i
i i
where d s s
i
jij
if s s
i i
i
Next we de ne a shift map on by
s s s s s s s i e s s
n n i i
The shift map is continuous and it has two xed p oints consisting
of a string of all s or all s A p erio d n orbit of is written as
s s s s s where the overbar indicates that the symb olic
n n
sequence rep eats forever A few of the p erio dic orbits and their shift
equivalent representations are listed b elow
Perio d
Perio d
Perio d
and so on In addition to p erio dic orbits of arbitrarily high p erio d the
shift map also p ossesses an uncountable in nity of nonp erio dic orbits as
well as a dense orbit See Devaney or Wiggins for the details
In section we showed that the shift map on the space of one
sided symb ol sequences is top ologically semiconjugate to the quadratic
map A similar results holds for the shift map on the space of bi in nite
sequences and the horsesho e map namely there exists a homeomor
phism connecting the dynamics of f on and on such
that f
f
The corresp ondence b etween the shift map on and the horsesho e map
on the invariant set is pretty easy to see again for the mathematical
CHAPTER DYNAMICAL SYSTEMS THEORY
Figure Equivalence b etween the dynamics of the horsesho e map
on the unit square and the shift map on the bi in nite symb ol space
details see Wiggins The invariant set consists of an in nite inter
section of horizontal and vertical strips These intersection p oints are
lab eled by their symb olic itinerary and the horsesho e map carries one
point of the invariant set to another precisely by a shift map Consider
for instance the p erio d two orbit
to and back again This corresp onds The shift map sends
to an orbit of the horsesho e map that b ounces back and forth b etween
the p oints lab eled and in see Fig The top ological
conjugacy b etween and f allows us to immediately conclude that
like the shift map the horsesho e map has
a countable in nity of p erio dic orbits and all the p erio dic orbits
are hyp erb olic saddles
an uncountable in nity of nonp erio dic orbits
a dense orbit
The shift map and hence the horsesho e map exhibits sensitive de
p endence on initial conditions see section As stated ab ove it
SMALE HORSESHOE
p ossesses a dense orbit These two prop erties are generally taken as
de ning a chaotic set
From Horsesho es to Tangles
Forward and backward iterations of the horsesho e map generate the
lo cations of p erio dic p oints to a higher and higher precision That is by
iterating the horsesho e map we can sp ecify the lo cation of a p erio dic
orbit within a homo clinic tangle of the horsesho e to any degree of
accuracy For instance after one iteration weknow the approximate
lo cation of a p erio d two orbit It lies somewhere within the shaded
regions lab eled and in Figure c After two iterations we
know its p osition even b etter It lies somewhere within the shaded
regions lab eled and Fig c
The forward iterates of the horsesho e map pro duce a snake that
u
approaches the unstable manifold W of the p erio dic p oint The back
ward iterates pro duce another snake that approaches the stable mani
s
fold W of the p erio dic p oint at the origin Thus iterating a horsesho e
generates a tangle The relative lo cations of horizontal and vertical
branches of this tangle are the same as those that o ccur in a homo clinic
tangle with a horsesho e arising in a particular ow This is illustrated
in Figure We can name the branches of the tangle with the same
lab eling scheme we used for the horsesho e For a horsesho e the lab el
ing scheme is easy to see once we notice that b oth the horizontal and
vertical branches are lab eled according to the alternating binary tree
intro duced in section
The lab eling of the horizontal branches is determined by the symbols
s s s For instance the horizontal lab el for branch H can b e
determined by reading down the alternating binary tree as illustrated
in Figure A second alternating binary tree is used to determine
the lab eling for the vertical branches The lab eling for the branch V
is indicated in Figure The lab eling scheme for the horizontal and
vertical branches at rst app ears complicated However the branch
names are easy to write down once we realize that they can b e read directly from the alternating binary tree
CHAPTER DYNAMICAL SYSTEMS THEORY
Figure Homo clinic horsesho e tangle and the lab eling scheme for
horizontal and vertical branches from a pair of alternating binary trees
HYPERBOLICITY
Figure Examples of horsesho e like maps that generate hyp erb olic
invariant sets
Hyp erb olicity
The horsesho e map is just one of an in nity of p ossible return maps
chaotic forms that can b e successfully analyzed using symb olic dy
namics A few other p ossibilities are shown in Figure Eachdif
ferent return map generates a di erent homo clinic tangle but all these
tangles can b e dissected using symb olic dynamics All these maps are
similar to the horsesho e b ecause they are top ologically conjugate to an
appropriate symb ol space with a shift map All these maps p ossess an
invariantCantor set These invariant sets all p ossess a sp ecial prop
erty that ensures their successful analysis using symb olic dynamics
namely hyp erb olicity
Recall our de nition of a hyp erb olic p oint A xed p ointofamap
is hyp erb olic if none of the mo duli of its eigenvalues exactly equals
one The notion of a hyp erb olic invariant set is a generalization of a
hyp erb olic xed p oint Informally to de ne a hyp erb olic set we extend
this prop erty of no eigenvalues on the unit circle to each p oint of the
invariant set In other words there is no center manifold for any p oint
of the invariant set Technically a set arising in a di eomorphism
f R R is a hyp erb olic set if
s u
there exists a pair of tangent lines E x and E x for each x
which are preserved by Df x
CHAPTER DYNAMICAL SYSTEMS THEORY
s u
E x and E x vary smo othly with x
there is a constant such that kDf x w k kw k for all
u s
w E x and kDf x w k kw k for all w E x
A more complete mathematical discussion of hyp erb olicity can b e found
in Devaney or Wiggins A general mathematical theory for the
symb olic analysis of chaotic hyp erb olic invariant sets is describ ed by
Devaney and go es under the rubric of subshifts and transition
matrices
Like the horsesho e the chaotic hyp erb olic invariant sets encoun
tered in the mathematics literature are often chaotic rep ellers In phys
ical applications on the other hand we are more commonly faced with
the analysis of nonhyp erb olic chaotic attractors The extension of sym
b olic analysis from the mathematical hyp erb olic regime to the more
physical nonhyp erb olic regime is still an active research question
Lyapunov Characteristic Exp onent
In section we informally intro duced the Lyapunov exp onentas
a simple measure of sensitive dep endence on initial conditions i e
chaotic b ehavior The notion of a Lyapunov exp onent is a general
ization of the idea of an eigenvalue as a measure of the stabilityofa
xed p ointora characteristic exp onent as the measure of the sta
bility of a p erio dic orbit For a chaotic tra jectory it is not sensible to
examine the instantaneous eigenvalue of a tra jectory The next b est
quantity therefore is an eigenvalue averaged over the whole tra jectory
The idea of measuring the average stability of a tra jectory leads us to
the formal notion of a Lyapunov exp onent The Lyapunov exp onentis
b est de ned by lo oking at the evolution under a ow of the tangent
manifold That is sensitive dep endence on initial conditions is most
clearly stated as an observation ab out the evolution of vectors in the
tangent manifold rather than the evolution of tra jectories in the ow
of the original manifold M
The tangent manifold at a p oint x written as TM is the collection
x
of all tangentvectors of the manifold M at the p oint x The tangent
manifold is a linear vector space The collection of all tangent manifolds
LYAPUNOV CHARACTERISTIC EXPONENT
Figure Evolution of vectors in the tangent manifold under a ow
is called the tangent bund le For instance for a surface emb edded in R
the tangent manifold at each p oint of the manifold is a tangent plane
More generally if the original manifold is of dimension n then the tan
gent manifold is a linear vector space also of dimension n For further
background material on manifolds tangent manifolds and ows see
Arnold
The integral curves of a ow on a manifold provide a smo oth foli
ation of that manifold in the following manner A p oint x M go es
to the p oint x M under the ow see Fig Nowwemakea
t
key observation the tangentvectors w TM are also carried bythe
x
ow this is called a Lie dragging so that we can set up a unique
corresp ondence b etween the tangentvectors in TM and the tangent
x
vectors in TM Namely for each w TM there exists a unique
x
x
t
t
vector D w TM TheLyapunov characteristic exponent of a
t
x
ow is de ned as
kD x w k
t
ln x w lim
t
t kw k
That is the Lyapunovcharacteristic exp onent measures the average
growth rate of vectors in the tangent manifold The corresp onding
Lyapunovcharacteristic exp onentofamapis
n
x w lim ln k Df x w k
n n
Dynamical Systems Theory
where
n n
Df x Df f x Df x w
A ow is said to have sensitive dependence on initial conditions if the
Lyapunovcharacteristic exp onent is p ositive From a physical p ointof
view the Lyapunov exp onentisavery useful indicator distinguishing
achaotic from a nonchaotic tra jectory
References and Notes
Go o d general references for the material in this chapter are listed here For
ordinary di erential equations see V I Arnold Ordinary di erential equa
tions MIT Press Cambridge MA Susp ensions are discussed in Z
Nitecki Di erentiable Dynamics MIT Press Cambridge MA and
S Wiggins Introduction to applied nonlinear dynamical systems and chaos
Springer Verlag New York The standard reference for applied dy
namical systems theory is J Guckenheimer and P Holmes Nonlinear oscil
lations dynamical systems and bifurcations of vector elds second printing
Springer Verlag New York
IHJM stands for Ikeda Hammel Jones and Moloney See S K Hammel C
K R T Jones and J V Moloney Global dynamical b ehavior of the optical
eld in a ring cavity J Opt So c Am B
The quotes are from Paul Blanchard Some of the intro ductory material
is based on notes from Paul Blanchard s Dynamical Systems Course
MA We are indebted to Paul and to Dick Hall for explaining to us the
chain recurrent p oint of view
C Conley Isolated Invariant Sets and the Morse Index CBMS Conference
Series J M Franks Homology and Dynamical Systems Conference
Board of the Mathematical Sciences Regional Conference Series in Mathematics
Numb er American Mathematical So ciety Providence R Easton
Isolating blo cks and epsilon chains for maps Physica D
J Marsden and A Tromba Vector Calculus third ed W H Freeman New
York
S Rasband Chaotic dynamics of nonlinear systems John Wiley New York
Chapter discusses xed p oint theory for maps
J Thompson and H Stewart Nonlinear dynamics and chaos John Wiley
New York
Problems
T Parker and L Chua Practical numerical algorithms for chaotic systems
Springer Verlag New York
A nice description of the homo clinic p erio dic orbit theorem of Birkho and
Smith is presented by E V Eschenazi Multistability and Basins of Attraction
in Driven Dynamical Systems Ph D Thesis Drexel University also see
section of R Abraham and C Shaw Dynamics The geometry of behavior
Part three Global behavior Aerial Press Santa Cruz CA
H G Solari and R Gilmore Relative rotation rates for driven dynamical
systems Phys Rev A L M Narducci and N B
Abraham Laser physics and laser instabilities World Scienti c New Jersey
S Smale The mathematics of time Essays on dynamical systems economic
processes and related topics Springer Verlag New York J Yorke and
K Alligo o d Perio d doubling cascades for attractors A prerequisite for horse
sho es Comm Math Phys
P Holmes Knotted p erio dic orbits in the susp ensions of Smale s horsesho e
Extended families and bifurcation sequences Physica D
R L Devaney Anintroduction to chaotic dynamical systems second edition
Addison Wesley New York
S Wiggins Introduction to applied nonlinear dynamical systems and chaos
Springer Verlag New York
P Cvitanovi c G H Gunaratne and I Pro caccia Top ological and metric
prop erties of H enon typ e strange attractors Phys Rev A
An elementary intro duction to Lyapunov exp onents is provided by S Souza
Machado R Rollins D Jacobs and J Hartman Studying chaotic systems
using micro computers and Lyapunov exp onents Am J Phys
For the state of the art in computing Lyapunov exp onents see P
Bryant R Brown and H Abarbanel Lyapunov exp onents from observed time
series Phys Rev Lett
Problems
Problems for section
Show that the H enon map with reduces to an equivalent form of the quadratic map
Dynamical Systems Theory
Intro duce the new variable t to the forced damp ed p endulum and
rewrite the equations for the vector eld so that the owis v S R
Write the IHJM optical map as a map of tworealvariables f x y R R
where z x iy Consider separately the cases where the parameters and
B are real and complex
Section
What is the limit set of a p oint x of a p erio d two orbit of the quadratic
map O x fx x g
i
Let f z S S with z e z Show that if is irrational then
p S
Give an example of a map f M M with L
Why is the lab el a go o d name for the nonwandering set
Section
Verify that the Jacobian for the H enon map is
x y
n n
x y
n n
Compute the divergence of the vector eld for the damp ed linear oscillator
x x x
Calculate the divergence of the vector eld for the Du ng equation forced
damp ed p endulum and mo dulated laser rate equation For what parameter
values are the rst two systems dissipative or conservative
Calculate the Jacobian for the quadratic map the sine circle map the Baker s
map and the IHJM optical map For what parameter valuesistheBaker s
map dissipative or conservative
The de nition of a dissipative dynamical system in section actually
includes expansive systems Howwould you rede ne the term dissipativeto
handle these cases separately
Solve the equation of rst variation for the vector eld F x v v x
x x v v
That is nd the time evolution of t t t t
x v x v Section
Problems
Explicitly derive the evolution op erator for the vector eld F x v v x
by solving the di erential equation for the harmonic oscillator in dimensionless
variables
Show that the following Jordan matrices have the indicated linear evo
lution op erators
a
t
e
tA
A e
t
e
b
cos t sin t
tA t
A e e
sin t cos t
c
tA t
A e e
t
Find an example of a di erential equation with an equilibrium p ointwhichis
linearly stable but not lo cally stable Hint See Wiggins reference
Find the xed p oints of the sine circle map and the Baker s map
Find all the xed p oints of the linear harmonic oscillatorx x x and
evaluate their lo cal stabilityfor and When are the xed p oints
asymptotically stable
Find the equilibrium p oints of the unforced Du ng equation and the damp ed
p endulum and analyze their linear stability
Section
u s
Calculate W and W for the planar vector eldx x y y x
Draw a picture from Figure to showthatiff x is lo cated at d then
the map is orientation reversing
Section
Follow section to construct the phase p ortrait for the laser rate equation
when a and b
Section
Calculate the inverse of the horsesho e map f describ ed in section
Dynamical Systems Theory
Chapter
Knots and Templates
Intro duction
Physicists are confronted with a fundamental challenge when studying
a nonlinear system to wit how are theory and exp erimenttobecom
pared for a chaotic system What prop erties of a chaotic system should
aphysical theory seek to explain or predict For a nonchaotic system
aphysical theory can attempt to predict the long term evolution of
an individual tra jectory Chaotic systems though exhibit sensitive
dep endence on initial conditions Long term predictability is not an
attainable or a sensible goal for a physical theory of chaos What is to
b e done
A consensus is now forming among physicists whichsays that a
physical theory for low dimensional chaotic systems should consist of
twointerlo cking comp onents
a qualitative enco ding of the top ological structure of the chaotic
attractor symb olic dynamics top ological invariants and
a quantitative description of the metric structure on the attractor
scaling functions transfer op erators fractal measures
Aphysicist s dynamical quest consists of rst dissecting the top olog
ical form of a strange set and second dressing this top ological form
with its metric structure
CHAPTER KNOTS AND TEMPLATES
In this chapter weintro duce one b eautiful approach to the rst
part of a physicist s dynamical quest that is unfolding the top ology
ofachaotic attractor This strategy takes advantage of geometrical
prop erties of chaotic attractors in phase space
Alow dimensional chaotic dynamical system with one unstable di
rection has a rich set of recurrence prop erties that are determined by
the unstable saddle p erio dic orbits emb edded within the strange set
These unstable p erio dic orbits provide a sort of skeleton on which
the strange attractor rests For ows in three dimensions these p e
rio dic orbits are closed curves or knots The knotting and linking of
these p erio dic orbits is a bifurcation invariant and hence can b e used
to identify or ngerprint a class of strange attractors Although a
chaotic system de es long term predictability itmay still p ossess go o d
short term predictability This short term predictability fundamentally
distinguishes low dimensional chaos from our notion of a random
pro cess Mindlin and co workers building on work initiated by
Solari and Gilmore recently develop ed this basic set of observa
tions into a coherentphysical theory for the top ological characterization
of chaotic sets arising in three dimensional ows The approach advo
cated by Mindlin Solari and Gilmore emphasizes the prominentrole
top ology must playinanyphysical theory of chaos In addition it sug
gests a useful approachtoward developing dynamical mo dels directly
from exp erimental time series
In recentyears the ergo dic theory of di erentiable dynamical sys
tems has played a prominent role in the description of chaotic physical
systems In particular algorithms have b een develop ed to compute
fractal dimensions metric entropies and Lyapunov exp onents
for a wide variety of exp erimental systems It is natural to consider
such ergo dic measures esp ecially if the ultimate aim is the characteri
zation of turbulent motions which are presumably of high dimension
However if the aim is simply to study and classify low dimensional
chaotic sets then top ological metho ds will certainly play an imp ortant
role Top ological signatures and ergo dic measures usually present dif
ferent asp ects of the same dynamical system though there are some
unifying principles b etween the two approaches which can often b e
found via symb olic dynamics The metric prop erties of a dynam
ical system are invariant under co ordinate transformations however
INTRODUCTION
they are not generally stable under bifurcations that o ccur during pa
rameter changes Top ological invariants on the other hand can b e
stable under parameter changes and therefore are useful in identifying
the same dynamical system at di erent parameter values
The aim of this chapter is to develop top ological metho ds suitable
for the classi cation and analysis of low dimensional nonlinear dynam
ical systems The techniques illustrated here are directly applicable to
a wide sp ectrum of exp eriments including lasers uid systems such
as those giving rise to surface waves the b ouncing ball system
describ ed in Chapter and the forced string vibrations describ ed in
Chapter
The ma jor device in this analysis is the template or knot holder
of the hyp erb olic chaotic limit set The template is a mathematical
construction rst intro duced by Birman and Williams and further
develop ed by Holmes and Williams Roughly atemplate is an
expanding map on a branched surface Templates are useful b ecause
p erio dic orbits from the owofachaotic hyp erb olic dynamical system
can b e placed on a template in suchaway as to preserve their original
top ological structure Thus templates provide a visualizable mo del for
the top ological organization of the limit set Templates can also b e
describ ed algebraically by nite matrices and this in turn gives us a
kind of homology theory for low dimensional chaotic limit sets
As recently describ ed by Mindlin Solari Natiello Gilmore and
Hou templates can b e reconstructed from a mo derate amountof
exp erimental data This reconstructed template can then b e used b oth
to classify the strange attractor and to make sp eci c predictions ab out
the p erio dic orbit structure of the underlying ow Strictly sp eaking
the template construction only applies to ows in the three sphere
S although it is hop ed that the basic metho dology illustrated by the
template theory can b e used to characterize ows in higher dimensions
The strategy b ehind the template theory is the following For a
nonlinear dynamical system there are generally two regimes that are
well understo o d the regime where a nite numb er of p erio dic orbits
exist and the hyp erb olic regime of fully develop ed chaos The essen
tial idea is to reconstruct the form of the fully develop ed chaotic limit
set from a non fully develop ed p ossibly nonhyp erb olic region in pa
rameter space Once the hyp erb olic limit set is identi ed then the
CHAPTER KNOTS AND TEMPLATES
top ological information gleaned from the hyp erb olic limit set can b e
used to make predictions ab out the chaotic limit set in other p ossibly
nonhyp erb olic parameter regimes since top ological invariants suchas
knot typ es linking numb ers and relative rotation rates are robust un
der parameter changes
In the next section we follow Auerbach and co workers to showhow
p erio dic orbits are extracted from an exp erimental time series
These p erio dic cycles are the primary exp erimental data that the tem
plate theory seeks to organize Section de nes the core mathemat
ical ideas we need from knot theory knots braids links Reidemeister
moves and invariants In section we describ e a simple but phys
ically useful top ological invariant called a relative rotation rate rst
intro duced by Solari and Gilmore Section discusses templates
their algebraic representation and their symb olic dynamics Here we
present a new algebraic description of templates in terms of framed
braids a representation suggested by Melvin This section also
shows how to calculate relative rotation rates from templates Section
provides examples of relative rotation rate calculations from two
common templates In section we apply the template theory to
the Du ng equation This nal section is directly applicable to ex
periments with nonlinear string motions such as those describ ed in
Chapter
Perio dic Orbit Extraction
Perio dic orbits are available in abundance from a single chaotic time
series To see why this is so consider a recurrent three dimensional ow
in the vicinityofahyp erb olic p erio dic orbit Fig Since the
ow is recurrent we can cho ose a surface of section in the vicinityof
this xed p oint This section gives a compact map of the disk onto
itself with at least one xed p oint In the vicinity of this xed p ointa
chaotic limit set a horsesho e containing an in nite numb er of unstable
p erio dic orbits can exist A single chaotic tra jectory meanders around
this chaotic limit set in an ergo dic manner passing arbitrarily close
to every p oint in the set including its starting p ointand each periodic
point
PERIODIC ORBIT EXTRACTION
Figure A recurrent ow around a hyp erb olic xed p oint
Figure a A close recurrence of a chaotic tra jectory b By gently
adjusting a segment of the chaotic tra jectory we can lo cate a nearby
p erio dic orbit Adapted from Cvitanovi c
Now let us consider a small segmentofthechaotic tra jectory that
returns close to some p erio dic p oint Intuitively we exp ect to b e able
to gently adjust the starting p oint of the segment so that the segment
precisely returns to its initial starting p oint thereby creating a p erio dic
orbit Fig
Based on these simple observations Auerbachandco workers
showed that the extraction of unstable p erio dic orbits from a chaotic
time series is surprisingly easy Very recently Lathrop and Kostelich
and Mindlin and co workers successfully applied this orbit
extraction technique to a strange attractor arising in an exp erimental
ow the Belousov Zhab otinskii chemical reaction
CHAPTER KNOTS AND TEMPLATES
As discussed in App endix H the idea of using the p erio dic orbits
of a nonlinear system to characterize the chaotic solutions go es backto
Poincar e In a sense Auerbach and co workers made the inverse
observation not only can p erio dic orbits b e used to describ e a chaotic
tra jectory buta chaotic tra jectory can also b e used to lo cate p erio dic
orbits
A real chaotic tra jectory of a strange attractor can b e viewed as
akindofrandomwalk on the unstable p erio dic orbits A segment
of a chaotic tra jectory approaches an unstable p erio dic orbit along its
stable manifold This approach can last for several cycles during which
time the system has go o d short term predictability Eventually though
the orbit is ejected along the unstable manifold and pro ceeds until it
is captured by the stable manifold of yet another p erio dic orbit A
convenient mathematical language to describ e this phenomenon is the
shadowing theory of Conley and Bowen Informally wesay that
a short segmentofa chaotic time series shadows an unstable p erio dic
orbit emb edded within the strange attractor This shadowing e ect
makes the unstable p erio dic orbits and hence the hyp erb olic limit
set observable In this way horsesho es and other hyp erb olic limit sets
can b e measured
This also suggests a simple test to distinguish low dimensional chaos
from noise Namely a time series from a chaotic pro cess should have
subsegments with strong recurrence prop erties In section we
give examples of recurrence plots that show these strong recurrence
prop erties These recurrence plots giveusa quick test for determining
if a time series is from a low dimensional chaotic pro cess
Algorithm
Let the vector x be a point on the strange attractor where x in our
i i
setting is three dimensional and the comp onents are either measured
directly from exp eriment e g x x x or created from an emb ed
n
ding x s s s of fs g the scalar time series data gen
i i i i i
i
erated from an exp erimental measurementofonevariable In the
three dimensional phase space the saddle orbits generally have a one
dimensional rep elling direction and a one dimensional attracting direc
tion When a tra jectory is near a saddle it approximately follows the
PERIODIC ORBIT EXTRACTION
motion of the p erio dic orbit For a data segment x x x near
i i i
a p erio dic orbit and an we can nd a smallest n suchthat
kx x k We will often write n kn In a forced system n is
i n i
simply the numb er of samples p er fundamental forcing p erio d and k is
the integer p erio d If the system is unforced then n can b e found by
constructing a histogram of the recurrence times as describ ed byLath
rop and Kostelich Candidates for p erio dic orbits emb edded in
the strange attractor are now given by all x which are k recurrent
i
p oints where k n n is the p erio d of the orbit
In practice we simply scan the time series for close returns strong
recurrence prop erties
kx x k
i n i
for k to nd the p erio d one p erio d two p erio d three orbits
and so on When the data are normalized to the maximum of the time
series then app ears to work well The recurrence criterion is
usually relaxed for higher order orbits and can b e made more stringent
for low order orbits
In a ow with a mo derate numb er of samples p er p erio d say n
the recurrent p oints tend to b e clustered ab out one another That
is if x is a recurrentpoint then x x are also likely to b e
i i i
recurrentpoints for the same p erio dic orbit simply b ecause the p eri
o dic orbit is b eing approached from its attracting direction We call
a cluster of such p oints a window The saddle orbit is estimated or
reconstructed bycho osing the orbit with the b est recurrence prop
erty minimum inawindow Alternatively the orbit can also b e
approximated byaveraging over nearby segments which is the more
appropriate pro cedure for maps
Data segments of strong recurrence in a chaotic time series are easily
seen in recurrenceplots which are obtained by plotting kx x k as a
i n i
function of i for xed n kn Di erent recurrence plots are necessary
for detecting p erio d one k orbits p erio d two k orbits and
so on Windows in these recurrence plots are clear signatures of near
periodicityover several complete p erio ds in the corresp onding segment
of the time series data The p erio dic orbits are reconstructed bycho os
ing the orbit at the b ottom of eachwindow Windows in recurrence
CHAPTER KNOTS AND TEMPLATES
plots also provide a quick test showing that the time series is not b eing
generated by noise but may b e coming from a low dimensional chaotic
pro cess
Lo cal Torsion
In addition to reconstructing the p erio dic orbits we can also extract
from a single chaotic time series the linearized ow in the vicinity of each
p erio dic orbit In particular it is p ossible to calculate the local torsion
that is howmuch the unstable manifold twists ab out the xed p oint
Let v b e the eigenvector corresp onding to the largest eigenvalue
u u
ab out the p erio dic orbit That is v is the lo cal linear approximation
u
of the unstable manifold of the saddle orbit To nd the lo cal torsion
we need to estimate the numb er of half twists v makes ab out the
u
p erio dic orbit The op erator S
T
Z
T
S exp DF d
T
gives the evolution of the variational vector This variational vector
will generate a strip under evolution and the numb er of half turns the
lo cal torsion is nothing but one half the linking numb er de ned in
section of the central line and the end of the strip de ned by
A fx t v t t T g
u
where x is the curve corresp onding to the p erio dic orbit
The evolution op erator can b e estimated from a time series by rst
noting that
Z
T
DF d exp
exp DF j t exp DF j t exp DF j t
T T T
n
Let x b e a recurrent p oint and let fx g b e a collection of p oints
i j
j
in some prede ned neighborhood about x ThenDF x isa
i i
matrix and we can use a least squares pro cedure on p oints from the
time series in the neighb orho o d of x to approximate b oth the Jaco
i
bian and the tangent map The lo cal torsion the number of
PERIODIC ORBIT EXTRACTION
half twists ab out the p erio dic orbit is a rather rough number calcu
lated from the pro duct of the Jacobians and hence is insensitive to the
numerical details of the approximation
Example Du ng Equation
To illustrate p erio dic orbit extraction we again consider the Du ng
oscillator in the following form
x x
x x x x f cos x
x
with the control parameters f and
At these parameter values a strange attractor exists but the
attractor is probably not hyp erb olic
The Du ng equation is numerically integrated for p erio ds with
steps p er p erio d Data are sampled and stored every steps so
that p oints are sampled p er p erio d A short segment of the chaotic
orbit from the strange attractor pro jected onto the x x phase space
is shown in Figure
Perio dic orbits are reconstructed from the sampled data using a
standard Euclidean metric The program used to extract the p erio dic
orbits of the Du ng oscillator is listed in App endix F The distances
d x x are plotted as a function of i for xed n kn with
i n i
n Samples of these recurrence plots are shown in Figure
for k andk The windows at the b ottom of these plots
are clear signatures of near p erio dicity in the corresp onding segment
of the time series data The segment of length n with the smallest
distance the b ottom of the window was chosen to represent the nearby
unstable p erio dic orbit Some of the unstable p erio dic orbits that were
reconstructed by this pro cedure are shown in Figure We b elieve
that the orbits shown in Figure correctly identify all the p erio d one
and some of the p erio d three orbits emb edded in the strange attractor
at these parameter values In addition a p erio d two orbit and higher
order p erio dic orbits can b e extracted see Table However this
may not b e all the p erio dic orbits in the system since some of them
CHAPTER KNOTS AND TEMPLATES
Figure Two dimensional pro jection of a data segment from the strange attractor of the Du ng oscillator
PERIODIC ORBIT EXTRACTION
Figure Recurrence plots Distances d x x plotted as a func
i n i
tion of i for xed n kn withn The windows in these plots
show near p erio dicityover at least a full p erio d in the corresp onding
segment of the time series data The b ottom of each window is a good
approximation to the nearby unstable p erio dic orbit a k b
k
mayhave basins of attraction separate from the basin of attraction for
the strange attractor
The p erio dic orbits give a spatial outline of the strange attractor
By sup erimp osing the orbits in Figure onto one another we recover
the picture of the strange attractor shown in the upp er left hand corner
of Figure Each p erio dic orbit indicated in Figure is actually a
closed curve in three space we only showatwo dimensional pro jection
in Fig
Nowwe make a simple but fundamental observation Eachofthese
p erio dic orbits is a knot and the p erio dic orbits of the strange attractor
are interwoven tied together in a very complicated pattern Our goal
in this chapter is to understand the knotting of these p erio dic orbits
and hence the spatial organization of the strange attractor We b egin
by reviewing a little knot theory
CHAPTER KNOTS AND TEMPLATES
Figure Some of the p erio dic orbits extracted from the chaotic
time series data of Figure a symmetric p erio d one orbit b c
asymmetric pair of p erio d one orbits d e symmetric p erio d three orbits f g asymmetric p erio d three orbits
KNOT THEORY
Figure Planar diagrams of knots a the trivial or unknot b
gure eight knot c left handed trefoil d right handed trefoil e
square knot f granny knot
Figure Link diagrams a Hopf link b Borromean rings c
Whitehead link
Knot Theory
Knot theory studies the placement of one dimensional ob jects called
strings in a three dimensional space A simple and accurate
picture of a knot is formed by taking a rop e and splicing the ends
together to form a closed curve A mathematician s knot is a non self
intersecting smo oth closed curve a string emb edded in three space A
two dimensional planar diagram of a knot is easy to draw As illustrated
in Figure we can pro ject a knot onto a plane using a solid broken
line to indicate an overcross undercross A collection of knots is called
a link Fig
The same knot can b e placed in space and drawn in planar dia
gram in an in nite numb er of di erentways The equivalence of two
di erent presentations of the same knot is usually very di cult to see
CHAPTER KNOTS AND TEMPLATES
Figure Equivalent planar diagrams of the trefoil knot
Figure Two knots whose equivalence is hard to demonstrate
Classi cation of knots and links is a fundamental problem in top ology
Given two separate knots or links wewould like to determine when
two knots are the same or di erent Two knots or links are said to
be topological ly equivalent if there exists a continuous transformation
carrying one knot or link into another That is we are allowed to
deform the knot in anyway without ever tearing or cutting the string
For instance Figure shows two top ologically equivalent planar di
agrams for the trefoil knot and a sequence of moves showing their
equivalence The two knots shown in Figure are also top ologically
equivalent However proving their equivalence by a sequence of moves
is a real challenge
A p erio dic orbit of a three dimensional ow is also a closed noninter
secting curve hence a knot A p erio dic orbit has a natural orientation
asso ciated with it the direction of time This leads us to study oriented
knots Formally an orientedknot is an emb edding S R where S
is oriented Informally anoriented knot is just a closed curve with an
arrow attached to it telling us the direction along the curve
KNOT THEORY
Figure Crossing conventions a p ositive b negative
The imp ortance of knot theory in the study of three dimensional
ows comes from the following observation The p erio dic orbits of a
three dimensional ow form a link In the chaotic regime this link
is extraordinarily complex consisting of an in nite numb er of p erio dic
orbits knots As the parameters of the owarevaried the comp onents
of this link the knots may collapse to p oints Hopf bifurcations or
coalesce saddle no de or pitchfork bifurcations But no component of
the link can intersect itself or any other component of the link b ecause
if it did then there would b e two di erent solutions based at the same
initial condition thus violating the uniqueness theorem for di erential
equations The linking of p erio dic orbits xes the top ological structure
of a three dimensional ow Moreover as weshowed in the previous
section p erio dic orbits and their linkings are directly available from
exp erimental data Thus knot theory is exp ected to playa key role in
anyphysical theory of three dimensional ows
Crossing Convention
In our study of p erio dic orbits wewillworkwithoriented knots and
links To each crossing C in an oriented knot or link we asso ciate a
sign C asshown in Figure Apositive cross also known as
right hand cross or overcross is written as C A negative
cross also known as a left hand cross or undercross is written as
C This de nition of crossing is the opp osite of the Artin
crossing convention adopted by Solari and Gilmore
CHAPTER KNOTS AND TEMPLATES
Figure Reidemeistermoves a untwist b pull apart c middle
slide
Reidemeister Moves
Reidemeister observed that two di erent planar diagrams of the same
knot represent top ologically equivalent knots under a sequence of just
three primary moves now called Reidemeister moves of typ e I I I and
I I I These Reidemeistermoves illustrated in Figure simplify the
study of knot equivalence byreducingittoatwo dimensional problem
The typeImove untwists a section of a string the typ e I I move pul ls
apart two strands and the typ e I I I move acts on three strings sliding the
middle strand b etween the outer strands The Reidemeister moves can
b e applied in an in nite numb er of combinations So knot equivalence
can still b e hard to show using only the Reidemeister moves
Invariants and Linking Numb ers
A more successful strategy for classifying knots and links involves the
construction of top ological invariants A topological invariant of a knot
or link is a quantity that do es not change under continuous deforma
tions of the strings The calculation of top ological invariants allows
us to bypass directly showing the geometric equivalence of two knots
since distinct knots must b e di erent if they disagree in at least one
top ological invariant What we really need of course is a complete
set of calculable top ological invariants This would allow us to de
nitely say when two knots or links are the same or di erent Unfor
tunately no complete set of calculable top ological invariants is known
KNOT THEORY
Figure Linking numb ers a one b zero
for knots However mathematicians have b een successful in develop
ing some very ne top ological invariants capable of distinguishing large
classes of knots
The linking numb er is a simple top ological invariant de ned for a
link on two oriented strings and Intuitively we exp ect the Hopf
link in Figure a to have linking numb er since the two strings
are linked once Similarly the two strings in Figure b are unlinked
and should have linking numb er The linking numb er which agrees
with this intuition is de ned by
X
C lk
C
In words we just add up the crossing numb ers for each cross b etween
the two strings and and divide bytwo The calculation of linking
numb ers is illustrated in Figure Note that the last example is
a planar diagram for the Whitehead link showing that links can b e
linked even when their linking numb er is zero
The linking number is an integer invariant More re ned algebraic
p olynomial invariantscanbede nedsuch as the Alexander p olynomial
and the Jones p olynomial We will not need these more re ned
invariants for our work here
Braid Group
Braid theory plays a fundamental role in knot theory since any oriented
link can b e represented by a closed braid Alexander s Theorem
The identi cation b etween links braids and the braid group allows
CHAPTER KNOTS AND TEMPLATES
Figure Examples of linking numb er calculations Adapted from
Kau man
us to pass back and forth b etween the geometric study of braids and
hence knots and links and the algebraic study of the braid group For
some problems the original geometric study of braids is useful For
many other problems a purely algebraic approachprovides the only
intelligible solution
A geometric braid is constructed b etween two horizontal level lines
with n base p oints chosen on the upp er and lower level lines From up
perbasepoints wedraw n strings or strands to the n lower base p oints
Fig a Note that the strands have a natural orientation from
top to b ottom The trivial braid is formed by taking the ith upp er base
point directly to the ith lower base p oint with no crossings b etween the
strands Fig b More typically some of the strands will inter
sect Wesay that the i st strand passes over the ith strand if there
is a p ositive crossing b etween the two strands see Crossing Conven
tion section As illustrated in Figure a an overcrossing
or right crossing b etween the i st and ith string is denoted bythe
symbol b Theinverse b represents an undercross or left cross i e
i
i
the i st strand go es under the ith strand Fig b
By connecting opp osite ends of the strands weforma closedbraid
KNOT THEORY
Figure a A braid on n strands b A trivial braid
Figure Braid op erators a b overcross b b undercross
i i
CHAPTER KNOTS AND TEMPLATES
Figure a Braid of a trefoil knot b Braid of a Hopf link
b Figure Braid on four strands whose braid word is b b
Each closed braid is equivalent to a knot or link and conversely Alexan
der s theorem states that any oriented link is represented by a closed
braid Figure a shows a closed braid on two strands that is equiv
alent to the trefoil knot Figure b shows a closed braid on three
strands that is equivalent to the Hopf link The representation of a
link by a closed braid is not unique However only two op erations on
braids the Markovmoves are needed to provetheidentitybetween
two top ologically equivalent braids
A general n braid can b e built up from successive applications of
the op erators b and b This construction is illustrated for a braid
i
i
on four strands in Figure The rst crossing b etween the second
and third strand is p ositive and is represented by the op erator b
The next crossing is negative b andisbetween the third and fourth
strands The last p ositive crossing is represented by the op erator b
Each geometric diagram for a braid is equivalent to an algebraic braid
KNOT THEORY
Figure Braid relations
word constructed from the op erators used to build the braid The braid
word for our example on four strands is b b b
Two imp ortant conventions are followed in constructing a braid
word First at each level of op eration b b onlytheith and i st
i
i
strands are involved There are no crossings b etween any other strands
atagiven level of op eration Second it is not the string but the base
p oint whichisnumb ered Each string involved in an op eration incre
ments or decrements its base p ointby one All other strings keep their
base p oints xed
The braid group on n strands B is de ned by the op erators fb i
n i
n g The identity elementof B is the trivial n braid How
n
ever as previously mentioned the expression of a braid group element
that is a braid word is not unique The top ological equivalence b e
tween seemingly di erent braid words is guaranteed by the braid rela
tions Fig
b b b b ji j j
i j j i
b b b b b b
i i i i i i
The braid relations are taken as the de ning relations of the braid
group Each top ologically equivalent class of braids represents a col
lection of words that are di erent representations for the same braid
in the braid group In principle the braid relations can b e used to
show the equivalence of anytwowords in this collection Finding a
practical solution to word equivalence is called the word problem The
word problem is the algebraic analog of the geometric braid equivalence problem
CHAPTER KNOTS AND TEMPLATES
Framed Braids
In our study of templates we will need to consider braids with fram
ing A framedbraid is a braid with a p ositive or negativeinteger
asso ciated to each strand This integer is called the framing We think
of the framing as representing an internal structure of the strand For
instance if each strand is a physical cable then we could sub ject this
cable to a torsional force causing a twist In this instance the framing
could represent the numb er of half twists in the cable
Geometrically a framed braid can b e represented bya ribbon graph
Take a braid diagram and replace each strand by a ribb on To see the
framing wetwist eachribbonbyaninteger numb er of half turns A
half twist is a rotation of the ribb on through radians A p ositive half
twist is a half twist with a p ositive crossing the rightmost half of the
ribb on crosses over the leftmost half of the ribb on Similarly a negative
half twist is a negative crossing of the ribb on Figure shows how
the framing is pictured as the numb er and direction of internal ribb on
crossings
This concludes our brief intro duction to knot theory Wenow turn
our attention to discussing how our rudimentary knowledge of knot
theory and knot invariants is used to characterize the p erio dic and
chaotic b ehavior of a three dimensional ow
Relative Rotation Rates
Solari and Gilmore intro duced the relative rotation rate in an
attempt to understand the organization of p erio dic orbits within a ow
The phase of a p erio dic orbit is de ned by the choice of a Poincar e
section Relative rotation rates make use of this choice of phase and
are top ological invariants that apply sp eci cally to p erio dic orbits in
three dimensional ows Our presentation of relative rotation rates
closely follows Eschenazi s
As usual we b egin with an example Figure shows a p erio d
two orbit a p erio d three orbit and their intersections with a surface of
section The relative rotation rate b etween an orbit pair is calculated
RELATIVE ROTATION RATES
Figure Geometric representation of a framed braid as a ribb on
graph The integer attached to each strand is the sum of the half twists
in the corresp onding branch of the ribb on graph
b eginning with the di erence vector formed at the surface of section
r x x y y
A B A B
where x y and x y are the co ordinates of the p erio dic orbits
A A B B
lab eled A of period p and B of p erio d p In general there will b e
A B
p p choices of initial conditions from whichtoform r at the surface
A B
of section To calculate the rotation rate consider the evolution of r
in p p p erio ds as it is carried along by the pair of p erio dic orbits
A B
The di erence vector r will makesomenumb er of rotations b efore
returning to its initial con guration This numb er of rotations divided
by p p is the relative rotation rate Essentially the relative rotation
A B
rate describ es the average numb er of rotations of the orbit A ab out the
orbit B or the orbit B ab out the orbit A In the example shown in
Figure the p erio d three orbit rotates around the p erio d two orbit
twice in six p erio ds or one third average rotations p er p erio d
The general de nition pro ceeds as follows Let A and B be two
orbits of p erio ds p and p that intersect the surface of section at
A B
CHAPTER KNOTS AND TEMPLATES
Figure Rotations b etween a p erio d two and p erio d three orbit
pair The rotation of the di erence vector b etween the two orbits is
calculated at the surface of section This vector is followed for
full p erio ds and the number of average rotations of the di erence vector
is the relative rotation rate b etween the two orbits Adapted from
Eschenazi
RELATIVE ROTATION RATES
The relative rotation rate R A B and b b b a a a
ij p p
B A
is
Z
d arctan r r R A B
ij
p p
A B
or in vector notation
Z
n r d r
R A B
ij
p p r r
A B
The integral extends over p p p erio ds and n is the unit vector
A B
orthogonal to the plane spanned bythevectors r and d r The
indices i and j denote the initial conditions a and b on the surface of
i j
section In the direction of the ow a clo ckwise rotation is p ositive
The self rotation rate R A A isalsowell de ned by equation
ij
if we establish the convention that R A A The rela
ii
tive rotation rate is clearly symmetric R A B R B A It also
ij ji
commonly o ccurs that di erent initial conditions give the same relative
rotation rates Further prop erties of relative rotation rates including
a discussion of their multiplicity have b een investigated bySolariand
Gilmore
Given a parameterization for the two p erio dic orbits their relative
rotation numb ers can b e calculated directly from equation by
numerical integration see App endix F There are however several
alternative metho ds for calculating R A B For instance imagine
ij
arranging the p erio dic orbit pair as a braid on two strands This is
illustrated in Figure where the orbits A and B are partitioned into
segments of length p and p each starting at a b and ending at
A B i j
a b Wekeep track of the crossings b etween A and B with the
i j
counter
ij
if A crosses over B from left to right
i j
if A crosses over B from right to left
i j
ij
if A do es not cross over B
i j
As previously mentioned the crossing convention and this de nition of the
relative rotation rate are the opp osite of those originally adopted by Solari and Gilmore
CHAPTER KNOTS AND TEMPLATES
Figure The orbit pair of Figure arranged as a braid on two
strands The relative rotation rates can b e computed bykeeping track
of all the crossings of the orbit A over the orbit B Each crossing adds
or subtracts a half twist to the rotation rate The linking number is
calculated from the sum of all the crossings of A over B Adapted
from Eschenazi
Then the relative rotation rates can b e computed from the formula
X
n p p R A B
i n j n A B ij
p p
A B
n
Using this same counter the linking number of knot theory is
X
lk A B i p and j p
ij A B
i j
The linking numb er is easily seen to b e the sum of the relative rotation
rates
X
lk A B R A B
ij
ij
An intertwining matrix is formed when the relative rotation rates for
all pairs of p erio dic orbits of a return map are collected in a p ossibly
in nite dimensional matrix Intertwining matrices have b een calcu
lated for several typ es of ows such as the susp ension of the Smale
horsesho e and the Du ng oscillator Perhaps the simplest way
to calculate intertwining matrices is from a template a calculation we
describ e in section
Intertwining matrices serve at least two imp ortant functions First
they help to predict bifurcation schemes and second they are used to
identify a return mapping mechanism
TEMPLATES
Again by uniqueness of solutions two orbits cannot interact through
a bifurcation unless all their relative rotation rates are identical with
resp ect to all other existing orbits With this simple observation in
mind a careful examination of the intertwining matrix often allows
us to make sp eci c predictions ab out the allowed and disallowed bi
furcation routes An intertwining matrix can give rise to bifurcation
selection rules i e it helps us to organize and understand orbit ge
nealogies A sp eci c example for a laser mo del is given in reference
Perhaps more imp ortantly intertwining matrices are used to iden
tify or ngerprint a return mapping mechanism As describ ed in sec
tion low order p erio dic orbits are easy to extract from b oth ex
p erimental chaotic time series and numerical simulations The relative
rotation rates of the extracted orbits can then b e arranged into an in
tertwining matrix and compared with known intertwining matrices to
identify the typ e of return map In essence intertwining matrices can
b e used as signatures for horsesho es and other typ es of hyp erb olic limit
sets
If the intertwining comes from the susp ension of a map then as
mentioned in section see Fig the intertwining matrix with
zero global torsion is usually presented as the standard matrix A
global torsion of adds a full twist to the susp ension of the return
map and this in turn adds additional crossings to each p erio dic orbit in
the susp ension If the global torsion is an integer GT then this integer
is added to eachelement of the standard intertwining matrix
Relative rotation rates can b e calculated from the symb olic dynam
ics of the return map or directly from a template if the return
map has a hyp erb olic regime We illustrate this latter calculation in
section
Templates
In section weprovide the mathematical background surrounding
template theory This initial section is mathematicallyadvanced Sec
tion contains a more pragmatic description of templates and can
b e read indep endently of section
CHAPTER KNOTS AND TEMPLATES
Before we b egin our description of templates we rst recall that the
dynamics on the attractor can havevery complex recurrence prop erties
due to the existence of homo clinic p oints see section Poincar e s
original observation ab out the complexity of systems with transverse
homo clinic intersections is stated in more mo dern terms as
Katok s Theorem Asmooth ow on a three manifold with p os
t
itive top ological entropy hasahyp erb olic closed orbit with a
transverse homo clinic p oint
Templates help to describ e the top ological organization of chaotic
ows in three manifolds In our description of templates wewillwork
mainly with forced systems so the phase space top ology is R S
However the use of templates works for any three dimensional ow
Perio dic orbits of a ow in a three manifold are smo oth closed curves
and are thus oriented knots Recall yet again that once a p erio dic orbit
is created say through a saddle no de or ip bifurcation its knot typ e
will not change as wemove through parameter space Changing the
knot typ e would imply self intersection and that violates uniqueness
of the solution Knot typ es along with linking numb ers and relative
rotation rates are top ological invariants that can b e used to predict
bifurcation schemes or to identify the dynamics b ehind a system
The p erio dic orbits can b e pro jected onto a plane and ar
ranged as a braid Strands of a braid can pass over or under one
another where our convention for p ositive and negative crossings is
given in section We next try to organize all the knot information
arising from a ow and this leads us to the notion of a template Our
informal description of templates follows the review article of Holmes
Motivation and Geometric Description
Before giving the general de nition of a template webeginby illus
trating how templates or knot holders can arise from a owinR In
accordance with Katok s theorem let O b e a closed hyp erb olic orbit
with a transversal intersection and a return map resembling a Smale
horsesho e Figure a For a sp eci c physical mo del the form of
the return map can b e obtained either bynumerical simulations or by
TEMPLATES
Figure a Susp ension of a Smale horsesho e typ e return map
for a system with a transversal homo clinic intersection This return
map resembles the orientation preserving H enon map b The dots
b oth solid and op en will b e the b oundary p oints of the branches in
u
a template The pieces of the unstable manifold W on the intervals
a a and b b generate tworibbons
analytical metho ds as describ ed by Wiggins The only p erio dic
orbit shown in Figure b is given by a solid dot The p oints
of transversal intersection indicated by op en dots are not p erio dic
orbits but according to the Smale Birkho homo clinic theorem
they are accumulation p oints for an in nite family of p erio dic orbits see
section
The p erio dic orbits in the susp ension of the horsesho e map have
a complex knotting and linking structure whichwas rst explored by
Birman and Williams using the template construction
Let us assume that our example is from a forced system Then
the simplest susp ension consistent with the horsesho e map is shown
in Figure a However this is not the only p ossible susp ension
We could put an arbitrary number of full twists around the homo clinic
orbit O The number of twists is called the global torsion and it is a
top ological invariantofthe ow In Figure b the susp ension of
the horsesho e with a global torsion of is illustrated by representing
CHAPTER KNOTS AND TEMPLATES
u
Figure Susp ension of intervals on W O a global torsion is
b global torsion is
u
the lift of the b oundaries on W O of the horsesho e as a braid of two
ribb ons
Note that adding a single twist adds one to the linking number
of the b oundaries of the horsesho e and this in turn adds one to the
relative rotation rates of all p erio dic orbits within the horsesho e That
is a change in the global torsion changes the linking and knot typ es
but it do es so in a systematic way In the horsesho e example the global
torsion is the relative rotation rate of the p erio d one orbits
To nish the template construction weidentify certain orbits in
the susp ension Heuristically we pro ject down along the stable mani
s u
folds W O onto the unstable manifold W O i e we collapse onto
u
W O For the Smale horsesho e this means that we rst identify
T
the ends of the ribb ons now called branches at in Figure a
T
and next identify and The resulting braid template for the
horsesho e is shown in Figure b The template itself maynow
b e deformed to several equivalent forms not necessarily resembling a
braid including the standard horsesho e template illustrated in Figure
For our purp oses a lift is a susp ension consisting of ow with a simple twist
TEMPLATES
Figure Template construction a identify the branch ends at
T u T
i e collapse onto W O and b identify and to get a
braid template
For this particular example it is easy to see that such a pro jection is
one to one on p erio dic orbits Eachpoint of the limit set has a distinct
symb ol sequence and thus lies on a distinct leaf of the stable manifold
s s
W O This pro jection takes each leaf of W O onto a distinct p oint
u
of W O In particular for each p erio dic p oint of the limit set there
u
is a unique p ointonW O
Each p erio dic orbit of the map corresp onds to some knot in the
u
template Since the collapse onto W O is one to one we can use the
standard symb olic names of the horsesho e map to name each knot in
the template sections Each knot will generate a symb olic
sequence of s and s indicated in Figure Conversely each p erio dic
symb olic sequence of s and s up to cyclic p ermutation will generate
a unique knot The three simplest p erio dic orbits and their symb olic
names are illustrated in Figure
If a template has more than two branches then wenumb er the
k branches of the template with the numb ers f k g In
this waywe asso ciate a symb ol from the set f k g to each
branch A p erio dic orbit of p erio d n on the template generates a se
quence of n symb ols from the set f k g as it passes through
the branches Conversely each p erio dic word up to cyclic p ermuta
CHAPTER KNOTS AND TEMPLATES
Figure Standard Smale horsesho e template Each p erio dic in
nite symb olic string of s and s generates a knot
tions generates a unique knot on the template The template itself
is not an invariant ob ject However from the template one can easily
calculate invariants suchasknottyp es and linking numb ers In this
sense it is a knot holder
The branches of a template are joined or glued at the branch
lines In a braid template all the branches are joined at the same
branch line Figure b is an example of a braid template and
Figure is an example of a nonbraided template holding the same
knots Forced systems always give rise to braid templates We will
work mostly with ful l braid templates i e templates which describ e
a full shift on k symb ols Franks and Williams haveshown that any
emb edded template can b e arranged via isotopy as a braid template
The template construction works for anyhyp erb olic ow in a three
manifold To accommo date the unforced situation we need the follow
ing more general de nition of a template
De nition A template is a branched surface T and a semi ow on
t
T such that the branched surface consists of the joining charts
and the splitting charts shown in Figure
TEMPLATES
Figure Some p erio dic orbits held by the horsesho e template Note
and are unlinked but the orbits and are linked that the orbits
once
The semi ow fails to b e a ow b ecause inverse orbits are not unique
at the branch lines In general the semi ow is an expanding map so
that some sections of the semi owmay also spill over at the branch
lines The prop erties that the template T are required to satisfy
t
are describ ed by the following theorem
Birman and Williams Template Theorem Given a ow
on a three manifold M having a hyp erb olic chain recurrentset
t
there is a template T T M such that the p erio dic orbits
t
under corresp ond with p erhaps a few sp eci ed exceptions
t
one to one to those under Onany nite subset of the p erio dic
t
orbits the corresp ondence can b e taken via isotopy
The corresp ondence is achieved by collapsing onto the strong stable
manifold Technically we establish the equivalence relation b etween
elements in the neighborhood N ofthechain recurrent set as follows
CHAPTER KNOTS AND TEMPLATES
Figure Template building charts a joining chart and b split
ting chart
x x if k x x k ast In other words x and
t t
x are equivalent if they lie in the same connected comp onent of some
lo cal stable manifold of a p oint x N
Orbits with the same asymptotic future are identi ed regardless of
their past By throwing out the history of a symb olic sequence we can
hop e to establish an ordering relation on the remaining symb ols and
thus develop a symb olic dynamics and kneading theory for templates
similar to that for one dimensional maps The symb olic dynamics of
orbits on templates as well as their kneading and bifurcation theory is
discussed in more detail in the excellent review article by Holmes
The few sp eci ed exceptions will b ecome clear when we consider
sp eci c examples In some instances it is necessary to create a few
p erio d one orbits in that do not actually exist in the original ow
t t
These virtual orbits can sometimes b e identi ed with p oints at in nity
in the chain recurrent set
Some examples of two branch templates that have arisen in physical
problems are shown in Figure the Smale horsesho e with global tor
sion and the Lorenz template and the Pirogon The Lorenz tem
plate is the rst knot holder originally studied by Birman and Williams
It describ es some of the knotting of orbits in the Lorenz equation
for thermal convection The horsesho e template describ es some of the
knots in the mo dulated laser rate equations mentioned in section
TEMPLATES
Figure Common two branch templates a Smale horsesho e with
global torsion b Lorenz ow showing equivalence to Lorenz mask
c Pirogon d Smale horsesho e with global torsion Adapted from Mindlin et al
CHAPTER KNOTS AND TEMPLATES
Figure Representation of a template as a framed braid using
standard insertion
Algebraic Description
In addition to the geometric view of a template it is useful to develop
an algebraic description Braid templates are describ ed by three pieces
of algebraic data The rst is a braid word describing the crossing
structure of the k branches of the template The second is the fram
ing describing the twisting in each branch that is the local or branch
torsion internal to each branch The third piece of data is the lay
ering information or insertion array which determines the order in
which branches are glued at the branch line Wenowdevelop some
conventions for drawing a geometric template In the pro cess we will
see that the rst piece of data the braid word actually contains the
last piece of data the insertion array Thus we conclude that a tem
plate is just an instance of a framed braid a braid word with framing
see sections
Drawing Conventions
A graph of a template consists of two parts a ribb on graph and a
layering or insertion graph Fig In the upp er section of a
template wedraw the ribb on graph which shows the intertwining of
the branches as well as the internal twisting lo cal torsion within each
TEMPLATES
Figure The layering graph can always b e moved to standard form
back to front
branch The lower section of the template shows the layering informa
tion that is the order in which branches are glued at the branchline
By convention we usually con ne the expanding part of the semi ow
on the template to the layering graph the branches of the layering
graph get wider b efore they are glued at the branch line We also often
draw the lo cal torsion as a series of half twists at the top of the ribb on
graph
In setting up the symb olic dynamics on the templates that is in
naming the knots we followtwo imp ortantconventions First at the
top of the ribb on graph we lab el eachofthek branches from left to right
withanumb er from the lab eling set i k Second from
nowonwe will always arrange the layering graph so that the branches
of the template are ordered back to front from left to right This second
convention is called the standard insertion
The standard insertion convention follows from the following ob
servation Anylayering graph can always b e isotop ed to the standard
form by a sequence of branchmoves that are liketyp e I I Reidemeister
moves This is illustrated in Figure where weshowalayering
graph in nonstandard form and a simple branchexchange that puts it
into standard form
The adoption of the standard insertion convention allows us to dis
p ense with the need to draw the layering graph The insertion infor
mation is now implicitly contained in the lower ordering left to right
CHAPTER KNOTS AND TEMPLATES
back to front of the template branches We see that the template is
well represented by a ribb on graph or a framed braid We will often
continue to drawthelayering graph However ifitisnotdrawn then
we are following the standard insertion convention A template with
standard insertion is a framed braid
These conventions and the framed braid representation of a tem
plate are illustrated in Figure for a series of two branch templates
Weshow the template its version following standard insertion and the
ribb on graph framed braid without the layering graph from whichwe
can write a braid word with framing We also show the braid linking
matrix for the template whichisintro duced in the next section
Braid Linking Matrix
A nice characterization of some of the linking data of the knots held
by a template is given by the braid linking matrix In particular in
the second half of section weshowhow to calculate the relative
rotation rates for all pairs of p erio dic orbits from the braid linking
matrix The braid linking matrix is a square symmetric k k matrix
de ned by
b the sum of half twists in the ith branch
ii
b the sum of the crossings b etween the
ij
B
ith and the j th branches of the ribb on graph
with standard insertion
The ith diagonal elementof B is the lo cal torsion of the ith branch
The o diagonal elements of B are twice the linking numb ers of the
ribb on graph for the ith and j th branches The braid linking matrix
describ es the linking of the branches within a template and is closely
related to the linking of the p erio d one orbits in the underlying ow
For the example shown in Figure the braid linking matrix is
B C
B
A
The braid linking matrix is equivalent to the orbit matrix and insertion array
previously intro duced by Mindlin et al See Melvin and Tu llaro for a pro of
TEMPLATES
Figure Examples of two branched templates their corresp onding
ribb on graphs framed braids with standard insertion and their braid linking matrices
CHAPTER KNOTS AND TEMPLATES
The braid linking matrix also allows us to compute how the strands
of the framed braid are p ermuted At the top of the ribb on graph
the branches of the template are ordered i k Atthe
b ottom of the ribb on graph each strand o ccupies some p ossibly new
p osition The new ordering or permutation of the strands is given
B
by
i i oddentries b with j i
B ij
o dd entries b with j i
ij
Informally to calculate the p ermutation on the ith strand we examine
the ith row of the braid linking matrix adding the number of odd
entries to the rightoftheith diagonal elementto i and subtracting
the numb er of o dd entries to the left
For example for the template shown in Figure we nd that
and The
B B B
permutation is That is the rst strand go es to the second
B
p osition and the third strand go es to the rst p osition The second
strand go es to the front third p osition
Lo cation of Knots
Given a knot on a template the symb olic name of the knot is deter
mined by recording the branches over which the knot passes Given a
template and a symb olic name howdowe draw the correct knot on a
template
There are two metho ds for nding the lo cation of a knot on a tem
plate with k branches The rst is global and consists of nding the lo
cations of all knots up to a length p erio d n by constructing the appro
priate k ary tree with n levels The second metho d known as knead
ing theory is lo cal Kneading theory is the more e cient metho d of
solution when we are dealing with just a few knots
Trees
A branch of a template is called orientation preserving if the lo cal tor
sion the numb er of half twists is an even integer Similarly a branch
is called orientation reversing if the lo cal torsion is an o dd integer A
convenientway to nd the relative lo cation of knots on a k branch
TEMPLATES
template is by constructing a k ary tree which enco des the ordering
of p oints on the orientation preserving and reversing branchesofthe
template
The ordering tree is de ned recursively as follows At the rst level
n we write the symb olic names for the branches from left to right
as k The second level n is constructed by recording
the symb olic names at the rst level of the tree according to the ordering
rule if the ith branch of the template is orientation preserving then
we write the branch names in forward order k if the
ith branch is orientation reversing then we write the symb olic names
at the rst level in reverse order k k The n st
level is constructed from the nth level by the same ordering rule if
the ith symb ol branch at the nth level is orientation preserving then
we record the ordering of the symb ols found at the nth level if the
ith symb ol lab els an orientation reversing branchthenwe reverse the
ordering of the symb ols found at the nth level
This rule is easier to use than to state The ordering rule is illus
trated in Figure for a three branch template Branch is orienta
tion reversing branches and are b oth orientation preserving Thus
we reverse the order of any branchatthen st level whose nth level
is lab eled with In this example we nd
and so on
To nd the ordering of the knots on the template we read down the
k ary tree recording the branch names through whichwe pass see Fig
The ordering at the nth level of the tree is the correct ordering
for all the knots of p erio d n on the template Returning to our example
we nd that the ordering up to p erio d twois
The symbol is read precedes and indicates
the ordering relation found from the ordering tree the order induced
by the template
CHAPTER KNOTS AND TEMPLATES
Figure Example lo cation of p erio d two knots n
TEMPLATES
Todraw the knots with the desired symb olic name on a template
we use the ordering found at the b ottom of the k ary tree Lastly
we draw connecting line segments b etween shift equivalent p erio dic
orbits as illustrated in Figure For instance the p erio d two orbit
is comp osed of two shift equivalent string segments and which
b elong to the branches and resp ectively
Kneading Theory
The limited version of kneading theory needed here is a simple rule
which allows us to determine the relative ordering of two or more orbits
on a template From an examination of the ordering tree we see that
the ordering relation b etween two itineraries s fs s s s g
i n
and s is given by s s if s s for i n and s s when the
i n
i n
number of symb ols in fs s g representing orientation reversing
n
branches is even or s s when the number of symb ols representing
n
n
orientation reversing branchesisodd
and on the template As an example consider the orbits
shown in Figure We rst construct all cyclic p ermutations of
these orbits
Next we sort these p ermutations in ascending order
Last we note that the only orientation reversing branchis sowe
need to reverse the ordering of the p oints and yielding
which agrees with the ordering shown on the ordering tree in Figure
Calculation of Relative Rotation Rates
Relative rotation rates can b e calculated from the symb olic dynamics
of the return map or directly from the template Wenow illustrate this
CHAPTER KNOTS AND TEMPLATES
latter calculation for the zero global torsion lift of the horsesho e We
then describ e a general algorithm for the calculation of relative rotation
rates from the symb olics
Horsesho e Example
Consider two p erio dic orbits A and B of p erio ds p and p Atthe
A B
surface of section a p erio dic orbit is lab eled by the set of initial condi
tions x x x each x corresp onding to some cyclic p ermutation
n i
of the symb olic name for the orbit That is it amounts to a choice
of phase for the p erio dic orbit For instance the p erio d three orbit
on the standard horsesho e template gives rise to three symb olic
names When calculating relative rotation rates it is
imp ortanttokeep track of this phase since di erentpermutations can
give rise to di erent relative rotation rates
To calculate the relative rotation rate b etween two p erio dic orbits
we rst represent each orbit by some symb olic name choice of phase
Next we form the comp osite template of length p p This is il
A B
lustrated for the p erio d three orbit and the p erio d one orbit
in Figure a The two p erio dic orbits can now b e extracted from
the comp osite template and presented as two strands of a pure braid
of length p p with the correct crossing data Figure b The
A B
self rotation rate is calculated in a similar way The case of the p erio d
two orbit in the horsesho e template is illustrated in Figure c d
General Algorithm
Although wehave illustrated this pro cess geometrically it is completely
algorithmic and algebraic For a general braid template the relative
ordering of the orbits at each branch line is determined from the sym
b olic names and kneading theory Given the ordering at the branch
lines and the form of the template all the crossings b etween orbits
are determined and hence so are the relative rotation rates Gilmore
develop ed a computer program that generates the full sp ectrum
of relative rotation rates when supplied with only the p erio dic orbit
matrix and insertion array for a de nition of p erio dic orbit matrix
also known as the template matrix see ref i e purely algebraic
TEMPLATES
Figure Relative rotation rates from the standard horsesho e tem
plate a comp osite template for the orbits and b the p e
rio dic orbits represented as pure braids c comp osite template for
calculating self rotation rate of d pure braid of and e the
intertwining matrix for the orbits and
CHAPTER KNOTS AND TEMPLATES
data Here we describ e an alternative algorithm based on the framed
braid presentation and the braid linking matrix This algorithm is im
plemented as a Mathematica package listed in App endix G
To calculate the relative rotation rate b etween two orbits we need
to keep track of three pieces of crossing data crossings b etween
two knots within the same branch recorded by the branch torsion
crossings b etween two knots on separate branches recorded bythe
branch crossings and any additional crossings o ccurring at the
insertion layer calculated from kneading theory One way to organize
this crossing information is illustrated in Figure whichshows the
braid linking matrix for a three branch template and twowords for
whichwe wish to calculate the relative rotation rate
Formulas can b e written down describing the relative rotation rate
calculation but we will instead try to describ e in words the rel
ative rotation rate arithmetic that is illustrated in Figure To
calculate the relative rotation rate b etween two orbits we use the fol
lowing sequence of steps
Write the braid linking matrix for the template with standard insertion re
arrange branches as necessary until you reachback to front form
In the row lab eled write the word w until the length of the row equals
the least common multiple LC M between the lengths of w and w do
the same with word w in row
Ab ove these rows create a new row called the zeroth level formed by the
braid linking matrix elements b where i indexes the rows
i i
Find all identical blocks of symb ols that is all places where the symb olics
in b oth words are identical these are b oxed in Figure Wrap around
from the end of the rows to the b eginning of the rows if appropriate
The remaining groups of symb ols are called unblocked regions for the un
blo cked regions write the zeroth level value mo d if even record a zero if
o dd record a one directly b elowtheword rows at the rst level
regions sum the zeroth level values ab ovetheblock i e add For the blo cked
up all the entries at the zeroth level that lie directly ab oveablock and write
this sum mo d at the second level
regions lo ok for a sign change orientation reversing For the unblo cked
branches from one pair of symb ols to the next i e but
i i iu iu
INTERTWINING MATRICES
or but and write a at the second level if there is a
i i iu iu
sign change or write a if there is no change of sign The counter u gives
the integer distance to the next unblo cked region Wrap around from the
end of the rows to the b eginning of the rows if necessary
Group all terms in the rst and second levels as indicated in Figure
Add all terms in each group mo d and write at the third level
Sum all the terms at the zeroth level and write the sum to the rightofthe
zeroth row
Sum all the terms at the third level and write the sum to the rightofthe
third row
To calculate the relative rotation rate add the sums of the zeroth level and
the third level and divide by the LC M found in step
The rules lo ok complicated but they can b e mastered in just a
few minutes after which time the calculation of relative rotation rates
b ecomes just an exercise in the rotation rate arithmetic
Intertwining Matrices
With the rules learned in the previous section we can now calculate rel
ative rotation rates and intertwining matrices directly from templates
For reference we present a few of these intertwining matrices b elow
Horsesho e
The intertwining matrix for the zero global torsion lift of the Smale
horsesho e is presented in Table
Lorenz
The intertwining matrix for the relative rotation rates from the Lorenz
template is presentedinTable Adding a global torsion of one a
full twist adds two to the braid linking matrix and it adds one to each
relative rotation rate i e eachentry of the intertwining matrix
CHAPTER KNOTS AND TEMPLATES
Figure Example of the relative rotation rate arithmetic
INTERTWINING MATRICES
Table Horsesho e intertwining matrix
Table Lorenz intertwining matrix
CHAPTER KNOTS AND TEMPLATES
Figure Template for the Du ng oscillator for the parameter
regime explored in section
Du ng Template
In this nal section we will apply the p erio dic orbit extraction tech
nique and the template theory to a chaotic time series from the Du ng
oscillator For the parameter regime discussed in section Gilmore
and Solari argued on theoretical grounds that the template for
the Du ng oscillator is that shown in Figure The resulting in
tertwining matrix up to p erio d three is presented in Table All
the relative rotation rates calculated from the p erio dic orbits extracted
from the chaotic time series agree see App endix G with those found
in Table whichwere calculated from the braid linking matrix for
the template shown in Figure
However not all orbits up to p erio d three were found in the chaotic
DUFFING TEMPLATE
Table Du ng intertwining matrix calculated from the template
for the Du ng oscillator All the p erio dic orbits were extracted from
a single chaotic time series except for those in italics the so called pruned orbits
Templates
and did not app ear time series In particular the orbits
to b e present Such orbits are said to b e pruned
The template theory helps to organize the p erio dic orbit structure
in the Du ng oscillator and other low dimensional chaotic pro cesses
Mindlin and co workers have carried the template theory further
than our discussion here In particular they showhow to extract not
only p erio dic orbits but also templates from a chaotic time series
Thus the template theory is a very promising rst step in the develop
ment of top ological mo dels of low dimensional chaotic pro cesses
References and Notes
G B Mindlin X J Hou H G Solari R Gilmore and N B Tu llaro Classi
cation of strange attractors byintegers Phys Rev Lett
G B Mindlin H G Solari M A Natiello R Gilmore and X J Hou Top o
logical analysis of chaotic time series data from the Belousov Zhab otinskii re
action J Nonlinear Sci
N B Tu llaro H Solari and R Gilmore Relative rotation rates Fingerprints
for strange attractors Phys Rev A
H G Solari and R Gilmore Relative rotation rates for driven dynamical
systems Phys Rev A
H G Solari and R Gilmore Organization of p erio dic orbits in the driven
Du ng oscillator Phys Rev A
J P Eckmann and D Ruelle Ergo dic theory of chaos and strange attractors
Rev Mo d Phys
P Grassb erger and I Pro caccia Measuring the strangeness of strange attrac
tors Physica D
T C Halsey M H Jensen L P Kadano I Pro caccia and B I Shraiman
Fractal measures and their singularities The characterization of strange sets
Phys Rev A erratum Phys Rev A
I Pro caccia The static and dynamic invariants that characterize chaos and
the relations b etween them in theory and exp eriments Physica Scripta T
References and Notes
J P Eckmann S O Kamphorst D Ruelle and S Cilib erto Lyapunovexpo
nents from time series Phys Rev A
J M Franks Homology and dynamical systems Conference Board of the
Mathematical Sciences Regional Conference Series in Mathematics Number
American Mathematical So ciety Providence
F Simonelli and J P Gollub Surface wavemodeinteractions E ects of sym
metry and degeneracy J Fluid Mech
J Birman and R Williams Knotted p erio dic orbits in dynamical systems I
Lorenz s equations Top ology J Birman and R Williams
Knotted p erio dic orbits in dynamical systems I I Knot holders for b ered knots
Contemp orary Mathematics
P Holmes Knots and orbit genealogies in nonlinear oscillators in New direc
tions in dynamical systems edited by T Bedford and J Swift London Math
ematical So ciety Lecture Notes Cambridge University Press Cambridge
pp P J Holmes and R F Williams Knotted p erio dic orbits
in susp ensions of Smale s horsesho e Torus knots and bifurcation sequences
Archives of Rational Mechanics Annals
D Auerbach P Cvitanovi c J P Eckmann G Gunaratne and I Pro caccia
Exploring chaotic motion through p erio dic orbits Phys Rev Lett
D P Lathrop and E J Kostelich Characterization of an exp erimental strange
attractor by p erio dic orbits Phys Rev A
P MelvinandN B Tu llaro Templates and framed braids Phys Rev A
R
N B Tu llaro Chaotic themes from strings Ph D Thesis Bryn Mawr College
P Cvitanovi c Chaos for cyclists in Noise and chaos in nonlinear dynamical
systems editedby F Moss Cambridge University Press Cambridge
H Poincar e Les m ethodes nouvel les de la m ecanique c eleste Vol
Gauthier Villars Paris reprinted byDover English translation
New metho ds of celestial mechanics NASA Technical Translations See
Volume I Section
R Easton Isolating blo cks and epsilon chains for maps Physica D
Knots and Templates
D L Gonz alez M O Magnasco G B Mindlin H Larrondo and L Romanelli
A universal departure from the classical p erio d doubling sp ectrum Physica
D
V F R Jones Knot theory and statistical mechanics Sci Am November
M Wadati T Deguchi and Y Akutsu Exactly solvable mo dels and knot
theory Phys Reps L H Kau man On knots
Princeton University Press Princeton NJ
E V Eschenazi Multistability and basins of attraction in driven dynamical
systems Ph D Thesis Drexel University
J Franks and R F Williams Entropy and knots Trans Am Math So c
S Wiggins Global bifurcations and chaos analytical methods Springer Verlag
New York
J Guckenheimer and P Holmes Nonlinear oscil lations dynamical systems and
bifurcation of vector elds Springer Verlag New York
R Gilmore Relative rotation rates from templates a Fortran program private
communication Address Dept of Physics Drexel University Philadel
phia PA
Problems
Problems for section
Calculate the linking numb ers for the links shown in Figures a and c
Cho ose di erentorientations for the knots and recalculate
Calculate the linking numb ers b etween the three orbits shown in Figure
Calculate the linking numb ers b etween the orbits and in Figure
Write the braid words for the braids shown in Figure
Verify that the braid group section is in fact a group
Find an equivalent braid to the braid shown in Figure and write down
its corresp onding braid word Use the braid relations to show the equivalence
of the two braids
Problems
Write down the braid words for the braids shown in Figure Use the
braid relations to demonstrate the equivalence of the braids as shown
Write down the braid word for the braid in Figure
Section
Calculate the relative rotation rates from b oth the A orbit and the B orbit in
Figure That is verify the equivalence of R A B and R B A inthis
instance Use the geometric metho d illustrated in Figure whichistaken
from Figure Attempt the same calculation directly from Figure
Show that the sum of the relative rotation rates is the linking numb er see
reference for more details
Show the equivalence of equations and
Section
Draw three di erent three branch templates and sketch their asso ciated return
maps Assume a linear expansive owoneachbranch Construct their braid
linking matrices
Show that the Lorenz template arises from the susp ension of a discontinuous
map Consider the evolution of a line segment connecting the two branches
at the middle
Draw the template in Figure as a ribb on graph and as a framed braid
What is its braid linking matrix
Verify that the strands of a braid are p ermuted according to equation
Verify the relative rotation rates in Figure e by the relative rotation
to the table rate arithmetic describ ed in section Add the orbit
Section
Calculate the intertwining matrix for the orbits shown in Figure up to
p erio d two
Section
Verify the rowinTable by the relative rotation arithmetic
App endix A
App endix A Bouncing Ball Co de
The dynamical state of the b ouncing ball system is sp eci ed by three variables
the ball s height y its velo city v and the time t The time also sp eci es the table s
vertical p osition s t Let t b e the time of the k th table ball collision and v the
k k
ball s velo cityimmediately after the k th collision The system evolves according to
the velocity and phase equations
v s t fv g t t s t g
k k k k k k
g t t s t s t v t t
k k k k k k k
which are collectively called the impact map The velo city equation states that
the relative ball table sp eed just after the k th collision is a fraction of its value
just b efore the k th collision The phase equation determines t given t by
k k
equating the table p osition and the ball p osition at time t t is the smallest
k k
strictly p ositive solution of the phase equation The simulation of the impact map
on a micro computer presents no real di culties The only somewhat subtle p oint
arises in nding an e ective algorithm for solving the phase equation whichisan
implicit algebraic equation in t
k
Wecho ose to solve for t bythebisection method b ecause of its great stability
k
and ease of co ding Other zero nding algorithms such as Newton s metho d are
not recommended b ecause of their sensitivity to initial starting values All bisection
metho ds must b e supplied with a natural step size for the metho d at hand The step
interval must b e large enough to work quickly yet small enough so as not to include
more than one zero on the interval Our solution to the problem is do cumented in
the function ndstep of the C program b elow In essence our step nding metho d
works as follows
If the relative impact velo city is large then t and t will not b e close so
k k
the step size need only b e some suitable fraction of the forcing p erio d On the
other hand if t and t are close then the step size needs to b e some fraction of
k k
the interval Weapproximate the interval b etween t and t by noting that the
k k
relativevelo citybetween the ball and the table always starts out p ositive and must
b e zero b efore they collide again Using the fact that the time b etween collisions is
small it is easy to showthat
A cos v
k k
k
A sin g
k
where is the phase of the previous impact and is the time it takes the relative
k k
velo citytoreach zero provides the correct order of magnitude for the step size
k
This algorithm is co ded in the following C program
For a discussion of the bisection metho d for nding the real zeros of a continuous
function see R W Hamming Introduction to applied numerical analysis McGraw
Hill New York p
Bouncing Ball Co de
bb c bouncing ball program
copyright by Nicholas B Tufillaro
Date written July
Date last modified August
This program simulates the dynamics of a bouncing ball subject to
repeated collisions with an oscillating wall
The INPUT is
delta v Amplitude Frequency damping cycles
where
delta the initial position of ball between and this is the
phase of forcing frequency that you start at phase mod PI
v the initial velocity of the ball this must be greater than
the initial velocity of the wall
A Amplitude of the oscillating wall
Freq Frequency of the oscillating wall in hertz
damp the impact parameter describing the energy loss per
collision No energy loss conservative case when d
maximum dissipation occurs at d
cycles the total length the simulation should be run in terms of
the number of forcing oscillations
Units CGS assumed
Compile with cc bb c lm O o bb
Bugs
include stdio h
include math h
CONSTANTS CGS Units
define STEPSPERCYCLE
define TOLERANCE e
define MAXITERATIONS
define PI
define G earth s gravitational constant
define STUCK
define EIGHTH
Macros
define max A B A B A B
define min A B A B A B
App endix A
Comments of variables
t is time since last impact
ti is time of last impact
tau is ti t time since start of simulation
xi is position of ball at last impact
vi is velocity of ball at last impact
w is velocity of wall
v is velocity of ball
Global Variables
double delta initial phase position of ball
double v initial ball velocity
double A table amplitude
double freq frequency of forcing
double damp impact parameter
double omega angular frequency PI f PI T
double T period of forcing
double cycles length of simulation
Functions
double s w x v d acc
step double find
double checkstep
double root
double fmod asin
main
f
int stuckcount
double t ti tau dt tstop xi vi tj xj vj
double t alpha t beta t ph variables for sticking case
read input parameters
scanf lf lf lf lf lf lf delta v A freq damp cycles
T freq
tstop T cycles
omega PI freq
delta delta freq
if v w f
printf Error Initial velocity less than wall velocity nn
printf Wall velocity g nn w
Bouncing Ball Co de
exit
g
t ti tau ti t xi s tau vi v
step ti xi vi dt find
t dt tau ti t stuckcount
while tau tstop f
t dt tau ti t
if impact tau t xi vi f
t root t dt t t xi vi ti
tj ti t
xj s tj
vj damp w tj damp v t vi
t xi xj vi vj ti tj
dt find step ti xi vi
if dt STUCK f sticking solution
if A omega omega G f
printf Stuck forever with tablenn
exit
g
if fabs G A omega omega
alpha omega asin G A omega omega t
else
t alpha T
beta T t alpha t
dt t beta t alpha STEPSPERCYCLE
t ph fmod ti delta T
if t ph t alpha tph t beta f
stuckcount
if t alpha t ph
tj T ti t alpha t ph
else
tj ti t alpha t ph
xj s tj vj w tj
t xi xj vi vj ti tj
g
if stuckcount f
printf g gnn fmod tj T v t vj
printf Ball Stuck Twicenn
exit
g
g
if tau tstop
App endix A
printf g gnn fmod tj T v t vj
dt checkstep dt ti xi vi
t dt tau ti dt
g
g
g
double s tau wall position
double tau
f
return A sin omega tau delta
g
double w tau wall velocity
double tau
f
return A omega cos omega tau delta
g
double acc tau table acceleration
double tau
f
return A omega omega sin omega tau delta
g
double x t xi vi ball position
double t xi vi
f
return xi vi t G t t
g
double v t vi ball velocity
double t vi
f
return vi G t
g
double d tau t xi vi distance between ball and wall
double tau t xi vi
f
return x t xi vi s tau
g
int impact tau t xi vi find when ball is below wall
Bouncing Ball Co de
double tau t xi vi
f
return d tau t xi vi
g
pick a good step for finding zero
step ti xi vi double find
double ti xi vi
f
double t max tstep
if vi w ti f should be alpha or sticking
return STUCK
g
if acc ti G
t max fabs w ti vi acc ti G
else
t max T STEPSPERCYCLE
if t max T TOLERANCE f
return STUCK
g
max T STEPSPERCYCLE tstep min EIGHTH t
return tstep
g
double root a b t xi vi ti
double a b
double t xi vi ti
f
double m
int count
count
while f
count
if count MAXITERATIONS f
printf ERROR infinite loop in rootnn
exit
g
if d ti a a xi vi d ti b b xi vi f
printf root finding error no zero on intervalnn
exit
g
m a b
if d ti m m xi vi jj b a T TOLERANCE
App endix A
return m
else if d ti a a xi vi d ti m m xi vi
b m
else
a m
g
g
double checkstep dt ti xi vi
double dt ti xi vi
f
int count
for count d ti dt dt xi vi count f
dt EIGHTH dt
if count f
printf Error Can t calculate dtnn
exit
g
g
return dt g
Cubic Oscillator
App endix B Exact Solutions for a Cubic Oscillator
The free conservative cubic oscillator equation arose in the single mo de
mo del of string vibrations and admits exact solutions in two sp ecial circumstances
The rst is the case of circular motion at a constant radius R In Figure we could
imagine circular orbits arising when the restoring force on the string just balances
the centrifugal force Plugging the ansatz
r R cos t R sin t B
into equation we see that it is indeed a solution provided the frequency is
adjusted to
KR B
c
The second solution app ears when we consider planar motion If all the motion
is con ned to the x z plane then the system is a single degree of freedom oscillator
whose equation of motion in the dimensionless form obtained from equation
is
x x x B
The exact solution to equation B is
x a a
K F arccos B
E a b a a b
where F is an elliptic integral of the rst kind and K F E is
the energy constant
E x x x B
and
p
E B b a
As in circular motion the frequency in planar motion is again shifted to a new value
given by
E
B
p
I a a b
The exact solutions for circular and planar motion are useful b enchmarks for testing
limiting cases of more general but not necessarily exact results
An exact solution for the more general case of an anharmonic oscillator
x a a x a x a x
is provided by Reynolds
App endix B
References and Notes
K Banerjee J Bhattacharjee and H Mani Classical anharmonic oscillators
Rescaling the p erturbation series Phys Rev A
For an elementary discussion of elliptic functions see App endix F of E A Jack
son Perspectives of nonlinear dynamics Vol Cambridge University Press
New York pp
M J Reynolds An exact solution in nonlinear oscillators J Phys A Math
Gen L L Also see N Euler W H Steeb and K Cyrus On
exact solutions for damp ed anharmonic oscillators J Phys A Math Gen L L
Ode
App endix C Ode Overview
Ode renders a numerical solution to the initial value problem for many families
of rst order di erential equations It is a programming language that resembles
the mathematical language so that the problem p osed by a system is easy to state
thereby making a numerical solution readily available Ode solves higher order
systems since a simple pro cedure converts an nth order equation into n rst order
equations Three distinct numerical metho ds are implemented at present Runge
Kutta Fehlb erg default Adams Moulton and Euler The Adams Moulton and
Runge Kutta routines are available with adaptive step size The Ode User s
Manual provides b oth a tutorial on applying Ode and a discussion of its design and
implementation
The user need only b e familiar with the fundamentals of the UNIX op erating
system to access and run this numerical software Ode provides
A simple problem oriented user interface
A table driven grammar simplifying extensions and changes to the language
A structure designed to ease the intro duction of new numerical metho ds and
Remarkable execution sp eed and capacity for large problems for an inter
pretive system
Ode currently runs on a wide range of micro computers and mainframes that supp ort
the UNIX op erating system Ode was develop ed at Reed College in the summer of
under a UNIX op erating system and is public domain software The program
is currently in use at numerous educational and industrial sites Reed College
Tektronix Inc U C Berkeley Bell Labs etc and the program and do cumentation
are available from some electronic networks suchasInternet
Ode solves the initial value problem for a family of rst order di erential equa
tions when provided with an explicit expression for each equation Ode parses a
set of equations initial conditions and control parameters and then provides an
e cientnumerical solution Ode makes the initial value problem easy to express
for example the Ode program
an ode to Euler
y
y y
print y from
step
prints
A UNIX Shell can b e used as a control language for Ode Indeed this allows
Ode to b e used in combination with other graphical or analytical to ols commonly
available with the UNIX op erating system For instance the following shel l script could b e used to generate a bifurcation diagram for the Du ng equation
App endix C
shell script using Ode to construct a bifurcation diagram
for the Duffing oscillator
Scan in F
Create a file bif data with the initial conditions
before running this shell script
forI in
do
forJ in
do
forK in
do
tail bif data lastline tmp
xo awk fprint g lastline tmp
vo awk fprint g lastline tmp
ode marker bif data
alpha
beta
gamma
F I J K
x xo
v vo
theta
x v
v alpha v x beta x F cos theta
theta gamma
F I J K t theta
print x v F every from PI
step PI PI
marker
done
done
done
In addition to Ode there exist many other packages that provide numerical
solutions to ordinary di erential equations However if you wish to write your own
routines see Chapter of Numerical Recipes
Ode
References and Notes
K Atkinson An introduction to numerical analysis John Wiley New York
Pages contain a review of the literature on the numerical solution
of ordinary di erential equations
N B Tu llaro and G A Ross Ode A program for the numerical solution of
ordinary di erential equations Bell Lab oratories Technical Memorandum TM
based on the Ode User s Manual Reed College Academic
Computer Center At last rep ort the latest version of the source co de
and do cumentation were lo cated on the REED VAX Contact the Reed College
Academic Computer Center at for more details
For a tutorial on the UNIX shell see S R Bourne The UNIX system Addison
Wesley Reading MA
W Press B Flannery S Teukolsky and W Vetterling Numerical recipes in C
Cambridge University Press New York
App endix D
App endix D Discrete Fourier Transform
The following C routine calculates a discrete Fourier transform and p ower sp ec
trum for a time series It is only meant as an illustrative example and will not b e
very useful for large data sets for which a fast Fourier transform is recommended
Discrete Fourier Transform and Power Spectrum
Calculates Power Spectrum from a Time Series
Copyright Nicholas B Tufillaro
include stdio h
include math h
define PI
define SIZE
double ts SIZE A SIZE B SIZE P SIZE
main
inti k p N L
double avg y sum psmax
read in and scale data points
i
while scanf lf y EOF
ts i y
i
get rid of last point and make sure
of data points is even
if i
i
else
i
L i N L
subtract out dc component from time series
for i avg i L i
avg ts i
avg avg L
now subtract out the mean value from the time series for i i L i
Discrete Fourier Transform
ts i ts i avg
o k guys ready to do Fourier transform
first do cosine series
for k k N k
for p sum p N p
sum ts p cos PI k p N
A k sum N
now do sine series
for k k N k
for p sum p N p
sum ts p sin PI k p N
B k sum N
lastly calculate the power spectrum
for i i N i
P i sqrt A i A i B i B i
find the maximum of the power spectrum to normalize
for i psmax i N i
if P i psmax
psmax P i
for i i N i
P i P i psmax
o k print out the results k P k
for k k N k
printf d g n k P k
App endix E
App endix E H enon s Trick
The numerical calculation of a Poincar e map from a cross section at rst app ears
to b e a rather tedious problem Consider for example calculating the Poincar e map
for an arbitrary three dimensional ow
dy dz dx
f x y z g x y z h x y z
dt dt dt
with a planar cross section
x y z
A simpleminded approach to this problem would involvenumerically integrating
the equations of motion until the cross section is pierced by the tra jectory An
intersection of the tra jectory with the cross section is determined by testing for a
change in sign of the variable in question Once an intersection is found a bisection
algorithm could b e used to hone in on the surface of section to any desired degree
of accuracy
H enon however suggested a very clever pro cedure that allows one to nd the
intersection p oint of the tra jectory and the cross section in one step Supp ose
that b etween the nth and the n st steps we ndachange of sign in the z
co ordinate
t x y z and t t x y z
n n n n n n n n
To nd the exact value in t at whichthe z co ordinate equals zero we can change
t from the indep endent to a dep endentvariable and change z from a dep endent
variable to the indep endentvariable This is accomplished by dividing the rst two
equations by dt dz and inverting the last equation
f x y z dy g x y z dt dx
dz h x y z dz h x y z dz h x y z
This new system can b e numerically integrated forward one step z z with
n
the initial values x y t A simple numerical integration metho d suchasa
n n n
Runge Kutta pro cedure is ideally suited for this single integration step
As an application of H enon s metho d consider the swinging Atwo o d s machine
SAM de ned by the conservative equations of motion
v ru cos
vu sin u
r
r v
u
H enon s Trick
A cross section for this system is de ned by r r Soweneedtochange the
indep endentvariable from t to When changes sign we can nd the Poincar e
map bynumerically integrating the equations
dv
ru cos
d u
du
vu sin
d ur
dr v
d u
dt
d u
Pictures of the resulting Poincar e map are found in reference
References and Notes
M H enon On the numerical computation of Poincar e maps Physica D
Also see Hao B L Elementary symbolic dynamics World Scienti c
New Jersey pp
N B Tu llaro Motions of a swinging Atwood s machine J Physique
App endix F
App endix F Perio dic Orbit Extraction Co de
The rst C routine b elowwas used to extract the p erio dic orbits from a chaotic
time series arising from the Du ng oscillator as describ ed in section This co de
is sp eci c to the Du ng oscillator and is provided b ecause it illustrates the co ding
techniques needed for p erio dic orbit extraction The second routine takes a pair
of extracted p erio dic orbits and calculates their relative rotation rates according to
equation The input to the relative rotation rate program is a pair of p erio dic
orbits plus a rst line containing some header information ab out the input le
fp c
Find all Periodic orbits of period P and tolerance TOL
Copyright by Nicholas B Tufillaro
Department of Physics
Bryn Mawr College Bryn Mawr PA USA
include stdio h
include math h
This Array Maximum must be greater then P STEPS
define AMAX
define DIST X X Y Y
float sqrt double X X X X Y Y Y Y
main ac av
char av
int m n cnt P STEPS LEN TWOLEN SHORTERPERIODICORBIT
float d ds phe AMAX x AMAX y AMAX TOL SHRTOL
float oldphe
process command line arguments P TOL
ac P atoi av ac TOL float atof av
set global variables
STEPS LEN P STEPS TWOLEN LEN
SHRTOL TOL
initialization get first LEN points
for n n LEN n
if scanf f f f phe n x n y n EOF
printf Not enough orbits for computations n
exit
Perio dic Orbit Extraction Co de
find periodic orbits
for
for n LEN n TWOLEN n
if scanf f f f phe n x n y n EOF
exit
for n n LEN n
d DIST x n x n LEN y n y n LEN
if d TOL
Identify shorter periodic orbits if any
This is a kludge
for m SHORTERPERIODICORBIT m P m
ds DIST x n x n m STEPS y n y n m STEPS
if ds SHRTOL
SHORTERPERIODICORBIT
End of kludge
if SHORTERPERIODICORBIT
if fabs phe n oldphe
oldphe phe n
printf n f f f n n float P float cnt
for m n m n LEN m
printf f f f n phe m x m y m
cnt
for n n LEN n
phe n phe n LEN x n x n LEN y n y n LEN
App endix F
rrr c
calculate Relative Rotation Rates of two periodic orbits
of periods PA PB
Copyright by Nicholas B Tufillaro
Department of Physics Bryn Mawr College
Bryn Mawr PA USA
INPUT
Data file with phase x and y listed for points in two
periodic orbits For the first input line of each
periodic orbit use for phase and give the period in x
For this first point y is ignored
include stdio h
include math h
define PI
define ARG X Y float atan double Y double X
main
int m n M N I J
float phe x y PA PB A B
inti j q Q
float rx ry RR
if scanf f f f phe x y EOF
printf Error empty input file n
exit
if phe
printf Error Input file does not begin with n
exit
if phe
PA x
for n m n
if scanf f f f phe x y EOF
printf Error not enough orbits n
exit
if phe break
Perio dic Orbit Extraction Co de
if fmod phe
I m n
m
A n phe A n y A n x
PB x
M n
for n m scanf f f f phe x y EOF n
if phe
printf Error Too many orbits n
exit
if fmod phe
J m n
m
B n phe B n y B n x
N n
Q int PA N
for i i PA i
for j j PB j
for m I i n J j q q Q q m n
if m m M
if n n N
if m M m
if n N n
rx q B n A m ry q B n A m
for q q Q q
if ARG rx q ry q PI ARG rx q ry q PI
ARG rx q ry q PI
ARG rx q ry q PI
RR i j PI ARG rx q ry q
ARG rx q ry q
else if ARG rx q ry q PI
ARG rx q ry q PI
ARG rx q ry q PI
ARG rx q ry q PI
RR i j ARG rx q ry q ARG rx q ry q PI
App endix F
else
RR i j ARG rx q ry q
ARG rx q ry q
printf nROTATION RATES n n
printf PA g PB g n n PA PB
for i i PA i
for j j PB j
printf Index d d Rotations g Rel Rot g n
i j RR i j PI RR i j PI PA PB
printf n
Relative Rotation Rate Package
App endix G Relative Rotation Rate Package
The following Mathematica package calculates relative rotation rates and inter
twining matrices by the metho ds describ ed in section
Relative Rotation Rates Mathematica Package
Date
Last Modified
Authors Copyright
A Lorentz N Tufillaro and P Melvin
Departments of Mathematics and Physics
Bryn Mawr College Bryn Mawr PA USA
Bugs
Many of these symbolic computations are exponential time algorithms
so they are slow for long periodic orbits and templates with many branches
A C version of these routines exists which runs considerably
faster The algorithm for the calculation of the relative
rotation rate from a single word pair however is polynomial time
About the Package
This collection of routines automates the process for the symbolic
calculation of relative rotation rates and the intertwining matrix
for an arbitrary template The template is represented algebraically
by a framed braid matrix which is specified by the global variable
bm in these routines This variable must be defined by the user
when these routines are entered
There are three major routines
RelRotRate word pair AllRelRotRates word pair and
Intertwine start row stop row
The input for the RelRotRate programs is a word pair which is just
a list of lists For example a valid input for these programs is
where the first word of the word pair is and the second word
is The input for the Intertwine program is just two integers
start row and stop row For instance Intertwine would produce
all relative rotation rates for all words of length between and In addition the routine
App endix G
GetFunCyc branches period
generates all fundamental cycles of length period for a template with
b branches or a symbolic alphabet of b letters This routine can
be useful for the cycle expansion techniques see R Artuso E Aurell
and P Cvitanovic Recycling of strange sets I and II Nonlinearity
Vol Num May p
References
G B Mindlin X J Hou H G Solari R Gilmore and N B Tufillaro
Classification of strange attractors by integers Phys Rev Lett
N B Tufillaro H G Solari and R Gilmore Relative rotation rates
fingerprints for strange attractors Phys Rev A
H G Solari and R Gilmore Organization of periodic orbits in the
driven Duffing oscillator Phys Rev A
H G Solari and R Gilmore Relative rotation rates for driven
dynamical systems Phys Rev A
The template braid matrix is a global variable that should be
defined by the user before this package is used Comment out the
default setting for the braid matrix
Examples for the braid matrix are presented below
Global Variable Abbreviation
bm braid matrix algebraic description of template
Braid Matrix Example The Horseshoe Template
bm
Braid Matrix Example Second Iterate of The Horseshoe Template
bm
Relative Rotation Rate Package
Calculate the relative rotation rate for a pair of words
Variables local to RelRotRate
wordp wordpair
lcm lowest common multiple of word pair lengths
ewordp expanded word pair
top top list
bottom bottom list
rrr relative rotation rate of word pair
RelRotRate wordp List
Block
lcm
ewordp top bottom
rrr
If wordp wordp Return Self rotation rate
ewordp ExpandWordPair wordp
top GetTop ewordp
bottom GetBottom ewordp
lcm Length top
rrr SumAll top SumAll bottom lcm
Return rrr
Permute the word pair list to generate
all possible relative rotation rates
AllRelRotRates wordp List
Block
i pa pb lcm wordstep rrr
firstword secondword rrrs
firstword wordp
secondword wordp
pa Length firstword pb Length secondword
App endix G
lcm LCM pa pb
wordstep lcm lcm pa pb
rrrs
For i i lcm i wordstep
rrr RelRotRate firstword secondword
rrrs Append rrrs firstword secondword rrr
firstword RotateRight firstword wordstep
Return rrrs
Calculate the Intertwining Matrix of all orbits of length startrow
to length stoprow startrow stoprow
Intertwine startrow Integer stoprow Integer
Block
b i j k rowsize rsum colsize csum mult
cycls rcycls ccycls rrr rrrs
b Length bm
rowsize rsum
For i startrow i stoprow i
cycls GetFunCyc b i rowsize Length cycls
rsum rowsize
For j j rowsize j
rcycls Append rcycls cycls j
colsize csum
For i i stoprow i
cycls GetFunCyc b i colsize Length cycls
csum colsize
For j j colsize j
ccycls Append ccycls cycls j
rrr rrrs
For i i rsum i
For j j csum j
rrr AllRelRotRates rcycls i ccycls j
mult Length rrr
rrrs rrr rrr For k k mult k
Relative Rotation Rate Package
rrrs Append rrrs rrr k
Print rrrs
If rcycls i ccycls j Break
Subroutines for major programs RelRotRate AllRelRotRates
Intertwine
ExpandWordPair wp List
Block
i lcm firstword secondword
firstword wp secondword wp
lcm LCM Length firstword Length secondword
Return
Flatten Table firstword i lcm Length firstword
Flatten Table secondword i lcm Length secondword
GetTop ewp List
Block
i lcm
lcm Length ewp
Return
Table bm ewp i ewp i i lcm
GetBottom ewp List
Block
i s j prevj nextj cnt lcm sgn
top ewpd b b b bot
lcm Length ewp
top GetTop ewp
ewpd ewp ewp
initialize rows b b b For i i lcm i
App endix G
AppendTo b AppendTo b AppendTo b
calculate b row
For i i lcm i
If ewpd i b i Mod top i
calculate b row
For s ewpd s s
For i i lcm i
j i s If j lcm j lcm
prevj j If prevj prevj lcm
If ewpd j
cnt
For k j ewpd k k
cnt cnt top k
If k lcm k
i
cnt Mod cnt
b prevj cnt
calculate b row
For i i lcm i
j i s If j lcm j lcm
nextj j If nextj lcm nextj lcm
k nextj
While ewpd nextj
nextj i
If nextj lcm nextj lcm
If Negative ewpd j ewpd nextj sgn sgn
b j sgn
bot Mod b b b
Return bot
SumAll l List
Relative Rotation Rate Package
Block i
Return Sum l i i Length l
Generates the fundamental cycles of length period for a template
with b branches
GetFunCyc b Integer period Integer
Block
cycles
get all cycles of length
cycles GetAllCyc b period period and b branches
cycles DelSubCyc b cycles delete nonfundamental subcycles
cycles DelCycPerm cycles delete cyclic permutations
of fundamental cycles
Return cycles
Subroutines for GetFunCyc
GetAllCyc branches Integer levels Integer
Block
n m i j
roots tree nextlevel cycle cycles
For i roots i branches i
roots Append roots i
creates full n ary tree recursively
tree roots
For n n levels n
nextlevel
For m m branches m
nextlevel Append nextlevel Last tree
tree Append tree Flatten nextlevel
reads up each branch of tree to root from left to right
cycles
For i i branches levels i
cycle For j levels j j
App endix G
k Ceiling i branches levels j
cycle Prepend cycle tree j k
cycles Append cycles cycle
Return cycles
DelSubCyc b Integer allcycls List
Block
i j k levels period numofdivs numofsub copies wordpos
cycles div word droplist
subcycles subword nonfun funcycls
Initializations and gets divisor list
cycles allcycls levels Length cycles
period levels div Divisors levels
Creates nonfundamental words from periodic orbits created from
divisor list
numofdivs Length div
For i i numofdivs i go throw divisor list
copies period div i
subcycles GetAllCyc b div i
numofsub Length subcycles
create subwords of lengths found in divisor list
For j subword j numofsub j
subword subcycles j
expand subwords to length of periodic orbits
For k word k copies k
word Flatten Append word subword
find positions of nonfundamental cycles in all cycles
wordpos Flatten Position cycles word
droplist Union Append droplist wordpos
this is a kludge to delete nonfundamental cycles
For i nonfun i Length droplist i
nonfun Append nonfun cycles droplist i
funcycls Complement cycles nonfun
Relative Rotation Rate Package
Return funcycls
DelCycPerm funcycls List
Block
size i j period
cycs word
cycs funcycls
size Length cycs period Length cycs
For i i size i
word cycs i
For j j period j
word RotateLeft word
cycs Complement cycs word
size size
Return cycs
App endix H
App endix H Historical Comments
How hard are the problems p osed by classical mechanics The b east in the ma
chine the b east wenow call chaos was discovered ab out a century ago bytheFrench
scientist and mathematician Henri Poincar e during the course of his investigations
on the three b o dy problem the great unsolved problem of classical mechanics
Poincar e in his magnum opus on the three b o dy problem New Methods of Celestial
Mechanics writes
When we try to represent the gure formed by these twocurves
and their intersections in a nite numb er eachofwhich corresp onds
to a doubly asymptotic solution these intersections form a typ e of
trellis tissue or grid with in nitely serrated mesh Neither of the two
curves must ever cut across itself again but it must b end backupon
itself in a very complex manner in order to cut across all of the meshes
in the grid an in nite number of times
The complexity of this gure will b e striking and I shall not even try
to draw it Nothing is more suitable for providing us with an idea of
the complex nature of the three b o dy problem and of all the problems
of dynamics in general where there is no uniform integral and where
the Bohlin series are divergent
Poincar e is describing his discovery of homo clinic solutions homo clinic intersec
tions or homo clinic tangles The existence of these homo clinic solutions solved
the three b o dy problem insofar as it proved that no solution of the typ e envisioned
by Jacobi or Hamilton could exist Volume I I I of Poincar e s New Methods of Celes
tial Mechanics from which the ab ove quote is taken is the foundational
work of mo dern dynamical systems theory
Poincar e s great theorem of celestial mechanics as stated in his prize winning
essay to the King of Sweden is the following
The canonical equations of celestial mechanics do not admit except
for some exceptional cases to b e discussed separately any analytical
and uniform integral b esides the energy integral
Simply put Poincar e s theorem says that there do es not exist a solution to the
three b o dy problem of the typ e assumed by the Hamilton Jacobi metho d or any
other metho d seeking an analytic solution to the di erential equations of motion
H Poincar e Les m ethodes nouvel les de la m ecanique c eleste Vol
Gauthier Villars Paris reprinted byDover English translation
New metho ds of celestial mechanics NASA Technical Translations
R Abraham and C Shaw Dynamics The geometry of behavior Vol
Aerial Press Santa Cruz CA
H Poincar e Sur le probl eme des trois corps et les equations de la dynamique
Acta Math
From Three Bo dies to Strange Sets
Poincar e solved the three b o dy problem byshowing that no solution exists at
least not of the typ e assumed by scientists of his era
But Poincar e did much more Confronted with his discovery of homo clinic
tangles Poincar ewentontoreinvent what is meantby a solution In the pro cess
Poincar e laid the foundations for several new branches of mathematics including
top ology ergo dic theory homology theory and the qualitative theory of di erential
equations Poincar e also p ointed out the p ossible uses of p erio dic orbits in taming
the b east
It seems at rst that this fact can b e of no interest what
ever for practice In fact there is a zero probability for the initial
conditions of the motion to b e precisely those corresp onding to a p e
rio dic solution However it can happ en that they di er very little
from them and this takes place precisely in the case where the old
metho ds are no longer applicable We can then advantageously take
the p erio dic solution as rst approximation as intermediate orbit to
use Gyld en s language
There is even more here is a fact whichIhave not b een able to demon
strate rigorously but which seems very probable to me nevertheless
Given equations of the form de ned in art and any particular
solution of these equations we can always nd a p erio dic solution
whose p erio d it is true is very long such that the di erence b etween
the two solutions is as small as we wish during as long a time as
we wish In addition these p erio dic solutions are so valuable for us
b ecause they are so to say the only breachbywhichwemay attempt
to enter an area heretofore deemed inaccessible
The p erio dic orbit theme is pursued in Chapter
Shortly thereafter Hadamard pro duced the rst example of an abstract
system exhibiting chaos the geo desics on a surface of constant negative curvature
Hadamard s example was later generalized by Anosov and is still one of the b est
mathematical examples of chaos in its most extreme form
In America George David Birkho continued in the wayofPoincar e He ex
amined the use of maps instead of ows and b egan the pro cess of hunting and
F Browder The mathematical heritage of Henri Poincar e Vol Pro c Sym
in Pure Math Vol American Mathematical So ciety Providence RI
R MacKay and J Meiss eds Hamiltonian dynamical systems Adam Hilger
Philadelphia
J Hadamard Les surfacesa curbures opp os es et leurs lignes g eo d esiques Journ
de Math
G D Birkho Surface transformations and their dynamical applications Acta
Math
App endix H
naming di erent critters that he called limit sets of the alpha and omega variety
Still without the aid of the computer Birkho was often fo oled by the b east into
thinking that nonintegrability implies complete ergo dicity It to ok the work of Kol
mogorov Arnold and Moser in the early s to show that the b east is more subtle
in its destructive tendencies and that it prefers to chew on rational frequencies
Few physicists were concerned ab out the b east or even aware of its existence
during the rst half of the twentieth century however this is more than understand
able when you consider that they had their hands full with quantum mechanics and
some less savory inventions Still there were some notable exceptions such as those
discussed by Brillouin in his b o ok Scienti c Uncertainty and Information
Near the end of the Second World War Cartwright and Littlewood and
Levinson showed that the b east liked to play the numb ers and could generate
solutions as random as a coin toss At rst they did not quite b elieve their results
however exp eriments with a simple circuit the van der Pol oscillator forced them
to accept the idea that a fully deterministic system can pro duce random results
what wenow think of as the de nition of chaos
In the young top ologist Steven Smale was sitting on the b eachinRio
playing with the b east when he rst saw that in its heart lay a horsesho e Smale
found that he could name the b east a hyp erb olic limit set even if he had trouble
seeing it since it was very very thin Moreover Smale found a simple wayto
unravel the horsesho e via symb olic dynamics Now that the b east was named
and dissected mathematicians had some interesting mathematical ob jects to play
with and Smale and his friends were o and running
At ab out the same time the meteorologist Ed Lorenz a former student ofG D
Birkho discovered that the b east was not only in the heavens as evidenced in the
three b o dy problem but maywell b e around us all the time in the atmosphere
Whereas Smale showed us how to name the b east Lorenz allowed us to see the
b east with computer simulations and judiciously chosen mo dels Lorenz p ointed
G D Birkho Col lected mathematical works Vol Dover New York
J Moser Stable and random motions in dynamical systems Ann Math Studies
Princeton University Press Princeton NJ
L Brillouin Scienti c uncertainty and information Academic Press New
York
M L Cartwright and N Littlewo o d On nonlinear di erential equations of the
second order I J Lond Math So c
N Levinson A second order di erential equation with singular solutions Ann
Math
S Smale The mathematics of time Essays on dynamical systems economic
processes and related topics Springer Verlag New York
R L Devaney Anintroduction to chaotic dynamical systems second ed
Addison Wesley New York
E N Lorenz Deterministic nonp erio dic ow J Atmos Sci
From Three Bo dies to Strange Sets
out the role return maps can play in understanding real systems how ubiquitous
the b east really is and howwe can b ecome more familiar with the b east through
simple mo dels such as the logistic map
This two pronged approach of using abstract top ological metho ds on the one
hand and insightful computer exp eriments on the other is central to the metho d
ology employed when studying nonlinear dynamical systems More than anything
else it probably b est de nes the nonlinear dynamical systems metho d as it has
develop ed since Lorenz and Smale In fact J von Neumann help ed invent the elec
tronic computer mainly to solve and provide insightinto nonlinear equations In
in an article called On the principles of large scale computing machines he
wrote
Our present analytic metho ds seem unsuitable for the solution of the
imp ortant problems arising in connection with non linear partial dif
ferential equations and in fact with virtually all typ es of non linear
problems in pure mathematics
really e cient high sp eed computing devices may in the eld of
non linear partial di erential equations as well as in many other elds
which are now di cult or entirely denied of access provide us with
those heuristic hints which are needed in all parts of mathematics for
genuine progress
Thank heavens for the Martians
E N Lorenz The problem of deducing the climate from the governing equa
tions Tellus
App endix I
App endix I Pro jects
In this app endix weprovide a brief guide to the literature on some topics that
might b e suitable for an advanced undergraduate research pro ject
Acoustics
a Review article
W Lauterb orn and U Parlitz Metho ds of chaos physics and their
application to acoustics J Acoust So c Am
b Oscillations in gas columns
T Yazaki S Takashima and F Mizutani Complex quasip erio dic
and chaotic states observed in thermally induced oscillations of gas
columns Phys Rev Lett
c Wineglass
A French A study of wineglass acoustics Am J Phys
Biology
a Review article
L Olsen and H Degn Chaos in biological systems Quart Rev Bio
phys
b Brain waves
P Rapp T Bashore J Martinerie A Albano I Zimmerman and A
Mees Dynamics of brain electrical activity Brain Top ography
c Gene structure
H Je rey Chaos game representation of gene structure Nucleic Acids
Res
Chemistry
a Chemical clo cks
S Scott Clo cks and chaos in chemistry New Scientist December
E Mielczarek J Turner D Leiter and L Davis Chem
ical clo cks Exp erimental and theoretical mo dels of nonlinear b ehavior
Am J Phys
Electronics
a Analog simulation of laser rate equations
M James and F Moss Analog simulation of a p erio dically mo dulated
laser mo del J Opt So c Am B for more
details ab out the circuit contact F Moss Dept of Physics University of St Louis St Louis MO
Pro jects
b Circuits
J Lesurf Chaos on the circuit b oard New Scientist June
A Ro driguez Vazquez J Huertas A Rueda B Perez Verdu
and L Chua Chaos from switched capacitor circuits Discrete maps
Pro c IEEE
c Double scroll
T Matsumoto L Chua and M Komuro The double scroll bifurca
tions Circuit theory and applications T Mat
sumoto L Chua and M Komuro The double scroll IEEE Trans
on Circuits and Systems CAS L Chua M
Komuro and T Matsumoto The double scroll family IEEE Trans
on Circuits and Systems CAS T Weldon
An inductorless double scroll circuit Am J Phys
Hydro dynamics
a Dripping faucet
R Cahalan H Leidecker and G Cahalan Chaotic rhythms of a drip
ping faucet Computing in Physics Jul Aug R Shaw
The dripping faucet as a model chaotic system Aerial Press Santa
Cruz CA H Yep ez N Nuniez A Salas Brito C Vargas and
L Viente Chaos in a dripping faucet Eur J Phys
b Hele Shaw cell
J Nye H Lean and A Wright Interfaces and falling drops in Hele
Shaw cell Eur J Phys
c Lorenz lo op
S Do dd Chaos in a convection lo op Reed College Senior Thesis
M Gorman P Widmann and K Robbins Nonlinear dynam
ics of a convection lo op Physica D P Widmann
M Gorman and K Robbins Nonlinear dynamics of a convection lo op
II Physica D
d Surface waves
R Apfel Whisp ering waves in a wineglass Am J Phys
S Douady and S Fauve Pattern selection in Fara
day instability Europhys Lett J P Gollub
Spatiotemp oral chaos in interfacial waves To app ear in New per
spectives in turbulence edited by S Orszag and L Sirovich Springer
Verlag J Miles and D Henderson Parametrically forced surface
waves Annu Rev Fluid Mech V Nevolin Para
metric excitation of surface waves J Eng Phys USSR
App endix I
Mechanics
a Artistic mobiles
O Viet J Wesfreid and E Guyon Art cin etique et chaos m ecanique
Eur J Phys
b Bicycle stability
Y H ena Dynamical stability of the bicycle Eur J Phys
J Papadop oulos Bicycling handling exp eriments you can do
preprint Cornell University C Miller The physical anatomy
of steering stability BikeTech Octob er
c Compass in a B eld
H Meissner and G Schmidt A simple exp eriment for studying the
transition from order to chaos Am J Phys
V Cro quette and C Poitou Cascade of p erio d doubling bifurcations
and large sto chasticity in the motions of a compass J Physique Lettres
L L
d Comp ound p endulum
N Pedersen and O So erensen The comp ound p endulum in interme
diate lab oratories and demonstrations Am J Phys
e Coupled p endula
K Naka jima T Yamashita and Y Ono dera Mechanical analogue of
active Josephson transmission line J Appl Phys
f Elastic p endulum
E Breitenb erger and R Mueller The elastic p endulum A nonlinear
paradigm J Math Phys M Olsson Why
do es a mass on a spring sometimes misb ehave Am J Phys
H Lai On the recurrence phenomenon of a reso
nant spring p endulum Am J Phys L Falk
Recurrence e ects in the parametric spring p endulum Am J Phys
M Rusbridge Motion of the sprung p endu
lum Am J Phys T Cayton The lab oratory
spring mass oscillator An example of parametric instability Am J
Phys J Lipham and V Pollak Constructing
a misb ehaving spring Am J Phys
g Forced b eam
See App endix C of F Mo on Chaotic vibrations John Wiley Sons
New York
h Forced p endulum
R Leven B Pomp e C Wilke and B Ko ch Exp eriments on p erio dic
and chaotic motions of a parametrically forced p endulum Physica D
Pro jects
i Impact oscillator
H Isomaki J Von Bo ehm and R Raty Devil s attractors and chaos
of a driven impact oscillator Phys Lett A
A Brahic Numerical study of a simple dynamical system Astron
Astrophys C N Bapat and C Bapat Impact pair
under p erio dic excitation J Sound Vib
j Impact p endulum
D Mo ore and S Shaw The exp erimental resp onse of an impacting
p endulum system Int J Nonlinear Mech S
Shaw and R Rand The transition to chaos in a simple mechanical
system Int J Nonlinear Mech
k Impact printer
P Tung and S Shaw The dynamics of an impact print hammer J
of Vibration Acoustics Stress and Reliability in Design
April
l Swinging Atwo o d s machine
J Casasayas A Nunes and N B Tu llaro Swinging Atwood s ma
chine Integrability and dynamics J de Physique
J Casasayas N B Tu llaro and A Nunes In nity manifold of a
swinging Atwo o d s machine Eur J Phys
N B Tu llaro A Nunes and J Casasayas Unb ounded orbits of a
swinging Atwood s machine Am J Phys
N B Tu llaro Integrable motion of a swinging Atwood s machine
Am J Phys N B Tu llaro Motions of a
swinging Atwo o d s machine J de Physique N
B Tu llaro Collision orbits of a swinging Atwo o d s machine J de
Physique N B Tu llaro T A Abb ott and D
J Gri ths Swinging Atwo o d s machine Am J Phys
B Bruhn Chaos and order in weakly coupled systems of
nonlinear oscillators Physica Scripta
m Swinging track
J Long The nonlinear e ects of a ball rolling in a swinging quarter
circle track Reed College Senior Thesis R Benenson and B
Marsh Coupled oscillations of a ball and a curved track p endulum
Am J Phys
n Wedge
N Whelan D Go o dings and J Cannizzo Two balls in one dimension
with gravity Phys Rev A H Lehtihet and B
Miller Numerical study of a billiard in a gravitational eld Physica
D A Matulich and B Miller Gravity in one dimen
sion Stability of a three particle system Celest Mech
B Miller and K Ravishankar Sto chastic mo deling of a billiard
in a gravitational eld J Stat Phys
App endix I
Optics
a Laser instabilities
N B Abraham A new fo cus on laser instabilities and chaos Laser
Fo cus May
b Light absorbing uid
G Indeb etouw and T Zukowski Nonlinear optical e ects in absorbing
uids Some undergraduate exp eriments Eur J Phys
c Maser equations
D Kleinman The maser rate equations and spiking Bell System Tech
J July
d Semiconductor lasers
H Winful Y Chen and J Liu Frequency lo cking quasip erio dicity
and chaos in mo dulated self pulsing semiconductor lasers Appl Phys
Lett J Camparo The dio de laser in atomic
physics Contemp Phys
e Dio de pump ed YAGlaser
N B Abraham L Molter and G Alman Exp eriments with a dio de
pump ed Nd YAG laser private communication Address Department
of Physics Bryn Mawr College Bryn Mawr PA USA
Spatial Temp oral Chaos
a Capillary ripples
N B Tu llaro Order disorder transition in capillary ripples Phys
Rev Lett H Riecke Stable wave number
kinks in parametrically excited standing waves Europhys Lett
A B Ezerskii M I Rabinovitch V P Reutov and
I M Starobinets Spatiotemp oral chaos in the parametric excitations
of a capillary ripple Sov Phys JEPT W
Eisenmenger Dynamic prop erties of the surface tension of water and
aqueous solutions of surface active agents with standing capillary waves
in the frequency range from kc s to Mc s Acustica
b Cellular automata
J P Reilly and N B Tu llaro Worlds within worlds An intro duction
to cellular automata The Physics Teacher February
c Coupled lattice maps
R Kapral Pattern formation in two dimensional arrays of coupled
discrete time oscillators Phys Rev A
d Ferrohydro dynamics
E M Karp Pattern selection and pattern comp etition in a ferrohy dro dynamic system Reed College Senior Thesis
Pro jects
e Video feedback
J Crutch eld Space time dynamics in video feedback Physica D
V Golub ev M Rabinovitch V Talanov V Shklover
andV Yakhno Critical phenomena in inhomogeneous excitable media
JEPT Lett G Hausler G Seckmeyer and T
Weiss Chaos and co op eration in nonlinear pictorial feedback systems
Appl Opt
Index
b ouncing ball k recurrent p oint
C program limit set
impact map pseudo orbit
impact relation limit set
phase map p erio d three implies chaos
sticking solution
abstract dynamical system
strange attractor
alternating binary tree
system
Arnold V I
velo city map
Artin crossing convention
braid
asymptotic b ehavior
group
asymptotically stable
relations
attracting
word
set
braid linking matrix
attractor
branch line
identi cation
branch torsion
autonomous
C programming language
backward limit set
Cantor set
backward orbit
Cartwright and Littlewo o d
Baker s map
center
basin of attraction
center manifold
b eam
theorem
Bendixson Ivar
center space
bi in nite symb ol sequence
chain recurrent
bifurcation
chain recurrent set
diagram
chain rule
summary diagram
chaos
binary expansion
chaos vs noise
binary tree
chaotic
BINGO
attractor
Birkho G D
rep eller
Birman and Williams template theo
set
rem
circle map
bisection metho d
close return
Borromean rings
INDEX
p oint closure
solution co e cient of restitution
state co existing solutions
eventually xed conservative
eventually p erio dic p oint contraction
evolution op erator correlation dimension
linear correlation integral
expansion cross section
extended phase space crossing convention
damping co e cient
fast Fourier transform
delay time
Feigenbaum
derivative
Feigenbaum s delta
of a comp osite function
FFT
of a map
gure eight knot
deterministic
nite Fourier series
Devaney R L
rst return map
devil s staircase
rst order system di erential equa
di eomorphism
tions
di erential
xed p oint
di erential equations
ip bifurcation
rst order system of
ow
geometric de nition
forced damp ed p endulum
numerical solution of
forward limit set
di erentiator
forward orbit
disconnected set
Fourier series
discrete Fourier transform
Fourier transform
C program
C program
dissipative
fractal
divergence
fractal dimension
domain of attraction
framed braid
Du ng equation
framing
template
frequency sp ectrum
dynamical quest
fully develop ed chaos
dynamical systems theory
functional comp osition
fundamental frequency
eigenvalues real matrix
elastic collision
geometric braid
emb edded time series
geometric convergence
emb edded variable
geometric series
emb edding
global torsion
di erential
granny knot
time delayed
graphical iteration of a one dimensional
equation of rst variation
map equilibrium
INDEX
Hadamard J Jacobian
half twist
Katok s theorem
harmonic balance
kneading theory
harmonic oscillator
knot
Hausdor F
problem
Heaviside function
typ es
H enon map
Jacobian
laser rate equation
H enon Michel
Levinson N
H enon s trick
Li and Yorke
hetero clinic
lift
point
limit cycle
tangle
linear approximation
Holmes P
linear di erential equation
homeomorphism
linear map
homo clinic
linear resonance
intersection
linear stability
point
linear system
solution
linearization
tangle
link
Hopf bifurcation
linking numb er
Hopf link
lo cal stability
horsesho e
lo cal torsion
intertwining matrix
logistic map
linear map
longitudinal vibrations
template
Lorentz force law
hyp erb olic
Lorenz equations
xed p oint
divergence
point
intertwining matrix
set
template
hyp erb olicity
Lorenz E N
hysteresis
Lyapunov exp onent
IHJM optical map
main resonance
incommensurate
Mandelbrot B
inelastic collision
manifold
initial condition
map
inset
maps to
integral curve
Martians
integration trick
Mathematica
intertwining matrix
matrix of partial derivatives
invariant set
metric
invertible map
mo d op erator
itinerary mo dulated laser equations
INDEX
Pirogon template multifractal
PLL
negative cross
Poincar e map
neutral stability
Poincar e section exp erimental
nonautonomous
Poincar e Jules Henri
noninvertible map
nonlinear dynamics
Poincar e Bendixson theorem
nonwandering
point attractor
point
p ositive cross
set
power sp ectrum
numerical metho ds
preimage
calculation of a Poincar e map
primary resonance
solution of ordinary di erential equa
principle of sup erp osition
tions
pruning
Ode
quadratic map
orbit
bifurcation diagram
orbit stability diagram
bifurcation diagram program
ordering relation
C program
ordering tree
qualitative analysis
orientation
qualitative solution
orientation preserving
quasip erio dic motion
oriented knot
quasip erio dic route to chaos
outset
quasip erio dic solution
parab olic map
recurrence
parametric oscillation
criterion
partition
plot
p edagogical approach
recurrentpoint
p endulum
Reidemeister moves
p erfect set
relative rotation rate
p erio d
arithmetic
p erio d doubling
C program
bifurcation
calculation
route to chaos
Mathematica package
p erio d one orbit
rep eller
p erio dic orbit extraction
rep elling
C program
resonance
p erio dic p oint
resp onse curve
p erio dic solution
ribb on graph
phase plane
rotating wave approximation
phase space
rotation matrix
exp erimental reconstruction
rubb er sheet mo del
phase lo cked lo op
saddle piezo electric lm
INDEX
attractor saddle p oint
rep eller saddle no de bifurcation
set SAM
strange vs chaotic attractor Sarkovskii s theorem
stretching and folding secondary resonance
strob oscopic map self rotation rate
subharmonic self similarity
susp ension semiconjugate
swinging Atwo o d s machine semideterministic
symb ol space semi ow
symb olic sensitive dep endence
co ordinate
dynamics separatrix
future sequence space
itinerary shadowing
shift equivalent
tangent bifurcation
shift map
tangent bundle
sideband
tangent manifold
Simple Theorem of Fermat
tangle
sine circle map
Taylor expansion
sink
template
slowly varying amplitude
tentmap
Smale horsesho e
time delay
Smale S
time series
Smale Birkho homo clinic theorem
time T map
solid torus
top ological chaos
solution curve
top ological conjugacy
source
top ological entropy
sp ectral signature
top ological invariant
sp ectrum analyzer
top ology
square knot
torus attractor
stability
torus doubling route to chaos
diagram
tra jectory
in one dimensional maps
transcritical bifurcation
of a p erio dic p oint
transient
stable manifold
behavior
of a map
precursor
stable no de
solution
stable space
transverse vibrations
standard insertion
trapping region
standard map
tree
state
trefoil
static
triggering circuit strange
INDEX
ultraharmonic
ultrasubharmonic
unimo dal map
unit interval
universality theory
unknot
unstable manifold
of a map
unstable no de
unstable p erio dic orbit
unstable space
vector eld
Verhulst P F
von Neumann J
wandering p oint
Whitehead link
winding numb er z