Furman University

Electronic Journal of Undergraduate Mathematics

Volume

QUANTIFYING CHAOS IN DYNAMICAL SYSTEMS WITH

LYAPUNOV EXPONENTS

MICHAEL VAN OPSTALL

Abstract In this pap er we analyze the dynamics of a four dimensional me

chanical system which exhibits sensitive dep endence on initial conditions The

aim of the pap er is to intro duce the basic ideas of chaos theory while assuming

only a course in ordinary dierential equations as a prerequisite

Introduction

Dynamical systems in short are systems which exhibit change As such the

eld of dynamical systems is varied and rich Many dynamical systems can b e

mo deled by systems of dierential equations or discrete dierence equations Such

systems are called deterministic Examples of such systems include those of classical

mechanics Sensitive dependence on initial conditions is a phenomenon where slight

distance b etween the initial conditions of a system grows exp onentially Determin

istic dynamical systems that exhibit a sensitive dep endence on initial conditions

are known as chaoticManyphysical systems are chaotic from the driven simple

p endulum to the more complex system mo deled in this pap er

Dynamical systems are classied as discrete or continuous A discrete dynamical

system given by one or more dierence equations is one in which a function f

is iterated on an initial condition x The set of all p oints generated by iterating

f b eginning with x is known as the orbit of x under f Acontinuous system is

generally given by one or more dierential equations Continuous orbits are known

as trajectories

There are several diculties in working with chaotic systems Systems of dif

ferential equations that b elievechaotically are always nonlinear This nonlinearity

makes an analytic solution to these equations dicult In addition to the nonlinear

ity a continuous system which exhibits sensitive dep endence on initial conditions

must have dimension of at least three that is it must have three indep endent

variables The system discussed in the present pap er has degree four and hence

cannot b e easily visualized Despite these diculties the fundamental concepts of

the science are accessible to anyone who has taken a course in ordinary dierential

equations

OneDimensional Discrete Systems Despite their simple nature systems

in a single variable can b e used to mo del many things One go o d example is the

logistic map x x l x which is used as a simple mo del for p opulation

n n n

Received by the editors Septemb er

Mathematics Subject Classication D

Key words and phrases Chaos Dierential Equations Dynamical Systems

This pap er was written while the author was an undergraduate at Hop e College

MICHAEL VAN OPSTALL

Figure The logistic map with with an attracting xed

p oint left and another with it a rep elling xed p oint

growth Some features of dynamical systems are easiest to demonstrate in single

dimensional systems so a few are describ ed here

The orbit of the function is computed according to the relation x f x

n n

The logistic map describ ed ab ove is an example of a onedimensional discrete sys

tem Apoint where the functions value is unaected by further iteration ie

x x is called a xedpoint A xed p ointwhich is approached by orbits

n n

is known as an attractor and one from which orbits diverge is a rep eller Figure

represents a logistic map with an attracting xed p oint and a chaotic

logistic map with a rep elling xed p oint

One way to quantify chaotic b ehavior in a system is to measure the divergence

n

between orbits of twopoints with small initial separation Assume f is the nth

iteration of a function f Then for two dierent initial conditions x and x the

n n

separation b etween these orbits is given b y jf x f xj as a function of the

numb er of iterations If we assume that the separation of the tra jectories grows or

shrinks exp onentially wehave

n n n

jf x f xj e

and is called the Lyapunov exp onent

If wetake the initial separation between tra jectories to b e small we obtain

n n n

f x f x df

log log

n n dx

n n

Noting that x f x f x wenddf dx using the chain rule

n n

n

n

Y

df

 n  n  

f f f f f x f x

m

dx

m

QUANTIFYING CHAOS IN DYNAMICAL SYSTEMS WITH LYAPUNOV EXPONENTS

From this wearrive at our nal formula for the Lyapunov exp onent of a one

dimensional discrete system

n n

Y X

 

lim log f x lim log jf x j

m m

n n

n n

m m

This exp onent represents the average exp onential rate of divergence of nearby

orbits A zero exp onent implies linear divergence A p ositive exp onent indicates

sensitive dep endence on initial conditions as p oints initially close together will

diverge exp onentially along neighb oring tra jectories Negative exp onents are found

in systems where tra jectories converge so the initial separation b etween twopoints

will decrease in time

Noting that the formula for a singledimensional Lyapunov exp onent is simply

an average of the logarithm of the size of the derivative a formula for continuous

systems can b e obtained For a continuous system the mean b ecomes the exp ected



value of log jf xj and wehave the following formula for the Lyapunov exp onent

for a single dimensional continuous system

Z



f xlogjf xjdx

Higher Dimensional Systems For many real systems a singledimensional

mo del is inadequate Unfortunately along with a b etter multidimensional mo del

we gain more problems in calculating the Lyapunov exp onent of a system The

equation derived for single variable discrete systems do es not directly applyand

nonlinear dierential equations p ose problems as they are dicult or imp ossible

to solve Wemust often resort to numerical metho ds to solve these problems

Phase Space The phase space of a system is the ndimensional space in

which the p oints of an ndimensional system reside A graph of tra jectories in the

phase space is known as a phase diagramFor two dimensional systems the phase

space lies in the plane known as the phase plane and is easily visualized For

higher dimensional systems however the phase space is often pro jected into two

dimensions for easy viewing In our system describ ed b elow the four variables

dening the phase space were paired to pro duce two phase diagrams Atwo

dimensional phase diagram often plots the velo city of a b o dy against its p osition

Attractors In the onedimensional case p oints to which orbits converged

were known as attracting xed p oints The xed p oint is a sp ecial case of an

attractor In higher dimensional spaces tra jectories with small initial separation

are sometimes pulled together into a single tra jectory an attractor In these higher

dimensional systems these attractors can b e curves or surfaces An attractor in a

chaotic system is known as a strange attractor

Lyapunov Exp onents In an ndimensional dynamical system wehave n

Lyapunov exp onents Each represents the divergence of k volume k length

k

k area etc The sign of the Lyapunov exp onents indicates the b ehavior of

nearby tra jectories A negative exp onent indicates that neighb oring tra jectories

converge to the same tra jectory A p ositive exp onent indicates that neighb oring

tra jectories diverge When tra jectories diverge exp onentially a slight error in mea

surement of the initial p oint could b e catastrophic as the error grows exp onentially

as well If in equation is taken to b e the slight error in measuring a systems

MICHAEL VAN OPSTALL

Figure Left Convergence of tra jectories Center two

concentric circular tra jectories Right divergence of tra

jectories

state eventuallythiserrorgrows in accordance with the Lyapunov exp onent Fig

ure represents the three typ es of tra jectory b ehavior

Any measurementtaken has some error The Lyapunov exp onent aords us

a measure of how quickly this error grows If the Lyapunov exp onent is negative

error actually decreases Consider the damp ed p endulum a slight error in measure

ment do es not lead to a large overall error since the p endulum eventually comes to

rest We are primarily interested in systems where one or more of the Lyapunov

exp onents is p ositive In accordance with our informal denition of chaos b ehavior

of a system exhibiting sensitive dep endence on initial conditions we can dene a

chaotic system as one with at least one p ositiveLyapunov exp onent Predictability

in a system is lost here and measurement error grows exp onentially

Experimental Setup

Physical Setup The physical dynamical system we studied consisted of a

p endulum of length and mass m attached to a blo ckofmassM oscillating on

the end of a spring with spring constant k This apparatus w as forced with forcing

p

function f tA cos t A sin twhich is the motion of a camshaft

of length displaced A units from the axis of rotation driven with frequency

Sensors connected to a Realtime VAX recorded the p osition of the cart and angular

displacementofthependulumx and resp ectively

To obtain data for the carts velo city and the p endulums angular velo cityv and

resp ectively the data was generated by taking numerical derivatives actually

the slop e b etween neighb oring p oints A diagram of our system app ears in Figure

Phase Diagrams After recording data for dierent frequencies of forcing

twodimensional phase plots were pro duced for v vs x and vs The system

was chaotic at high driving frequencies The phase diagrams are given in Figure

Figure illustrates chaotic and p erio dic time series The problem of exp erimental

noise is quite evident in these gures Spurious data p oints can cause problems

when calculating the Lyapunov exp onents

Equations of Motion Equations of motion for this system can b e obtained

using the Lagrangian metho d The Lagrangian L is as the dierence b etween

QUANTIFYING CHAOS IN DYNAMICAL SYSTEMS WITH LYAPUNOV EXPONENTS

Figure Aschematic diagram of our physical system

Figure Phase plots of angular velo city vs angular p osition for

the p endulum The left is p erio dic and the right is a p ortion of the

chaotic phase diagram

potential and kinetic energy of a dynamical system The p osition r tofthecart

and r t of the p endulum are given by

r t x

r t x sin cos

The square of the velo cityofeach b o dy is calculated by taking the square of the

magnitude of the derivative of the p osition function



kr tk x



kr tk x x cos

From the p osition and velo city functions we get the Lagrangian

L M x mx x cos k x f mg cos

MICHAEL VAN OPSTALL

Figure Time series diagrams of angular p osition vs time The

left is p erio dic and the rightchaotic

where f t is the displacement of the forcing piston at time t The nal equations

d L L

of motion are derived according to the EulerLagrange equations

dt q q

k k

where t he q are the co ordinates of the system x and in our case These

k

equations yield the nal equations of motion for this system

M mx m sin m cos k x f

x cos g sin

These equations can b e further broken down into a system of four rstorder

dierential equations suitable for numerical integration

Calculating Lyapunov Exponents

Given a system of dierential equations numerical integration aords us a

metho d for determining the theoretical value for the systems Lyapunov exp onents

This metho d describ ed in detail byWolf et al in and describ ed b elow is use

ful for determining p ositiveLyapunov exp onents for chaotic systems The system

of equations of motion must b e converted to strictly rst order equations For an

ndimensional system n copies of n linearized equations are needed This lineariza

ymultiplying the Jacobian matrix of partial derivatives of tion is accomplished b

the n nonlinear functions by a column vector of the variables comparable to ap

proximating functions bya tangent line

Each of the linearized equations determines a p ointinn space with a separation

from the nonlinear system We start with a sphere of states centered on the non

linear tra jectory with linearized tra jectories tangent to the spheres surface This

sphere is really nothing more than an orthonormal frame of vectors but taking these

vectors to form a sphere aids in visualization of the pro cess The initial state vectors

dened in each direction are chosen to b e orthonormal each one p erp endicular to

the others and of unit length Now the system is allowed to evolveover time

After a short time the sphere of vectors has b ecome an ellipsoid with all vectors

approaching the direction of greatest growth

This presents a problem If this is allowed to continue indenitely all the vectors

will collapse onto the same vector and b ecome indistinguishable Additionallyif

QUANTIFYING CHAOS IN DYNAMICAL SYSTEMS WITH LYAPUNOV EXPONENTS

the largest vector continues to grow without limit it will so on approach the size of

the attractor which is the metric diameter of the set of p oints that makeupthe

attractor at which p oint the attractor folds backonto itself vectors whichhave

grown to o large collapse to small vectors causing a miscalculation we lose the fact

that the vector has grown and p erhaps note incorrectly that it has shrunk These

are the two ma jor problems in calculating Lyapunov exp onents Both are solved

simultaneously by renormalizing p erio dically using GramSchmidt orthonormaliza

tion After a sp ecied time the vectors are measured and are orthogonalized and

brought backtoavery small length The pro cess is rep eated after orthonormaliza

tion so an average can b e taken

The highest Lyapunov exp onent recall that an ndimensional system has n

Lyapunov exp onents can b e calculated by measuring the lengths of the largest

vector n times over a p erio d of t seconds where is the length of the vector at

n

measurement n Then the Lyapunov exp onentis

n

X

m

log

n

m

m

To nd the other exp onents one must monitor the evolution of area or hvolume

in the phase space Then using an analogous formula to the one ab overeplacing

the lengths with area or volume A The calculation then yields the sum of the

rst h exp onents where h is the numb er of dimensions of the space b eing measured

ie h represents length h represents area

Conclusions

The calculation of Lyapunov exp onents from collected data is similar to the

pro cess outlined ab ove for dierential equations but pitfalls ab ound Exp erimental

noise is an issue and phase space reconstruction see must b e considered for

systems when less than all of the phase variables can b e measured Since tra jectories

are built from exp erimental data and not equations no numerical integration is

required however See for greater detail on calculation of Lyapunov exp onents

Although many mo dern denitions of chaos are given in terms of orbits and p e

rio dicity the denition given in the present pap er is adequate for applications in

many systems of classical mechanics The aim of this pap er is to encourage other

pro jects Examples of other chaotic systems include the driven simple p endulum

the double p endulum or a multiple massspring system These systems can b e

constructed in most college or universityphysics labs We are grateful to Profes

sor Paul De Young from the Hop e College physics department for setting up our

physical system

In addition to observation of chaotic b ehavior such a pro ject reinforces or in

tro duces several other skills in ordinary dieren tial equations Mo deling using the

Lagrangian or Newtons laws can b e practiced in these more complex systems Ad

ditionally chaos is an excellentcontext in whichtointro duce nonlinear equations

and phase diagramming Indeed for simpler systems it is p ossible to involvea

computer algebra system such as Maple or Mathematica in drawing phase plots

References

GL Baker and JP Gollub Chaotic Dynamics AnIntroductionCambridge University Press

MICHAEL VAN OPSTALL

A Wolf JB Swift HL Swinney JA Vastano Determining Lyapunov Exp onents from a

Time Series Physica D

NH Packard JPCrutcheld JD Farmer and RS Shaw Geometry from a Time Series

Physical Review Letters

JC Roux RH Simoyi HL Swinney Observation of a Strange Attractor PhysicaD

HDI Abarbanel R Brown JJ Sidorowich LS Tsimring The Analysis of Observed

Chaotic Data in Physical Systems Reviews in Modern Physics

G Benettin L Galgani A Giorgelli JM Strelcyn Lyapunov Characteristic Exp onents for

Smo oth Dynamical Systems and for Hamiltonian Systems A Metho d for Computing all of

Them Meccanica

P Bugl Dierential Equations Matrices and ModelsPrentice Hall

Mathematics Department University of Washington Seattle Washington USA

Email address vnopstaluwashingtonedu

Sponsor Timothy Pennings Hope College Holland Michigan USA

Email address penningsmathhopeedu