Strong Chaos without Buttery Eect

in Dynamical Systems with Feedback

1 2 3

Guido Boetta Giovanni Paladin and Angelo Vulpiani

1

Istituto di Fisica Generale Universita di Torino

Via PGiuria I Torino Italy

2

Dipartimento di Fisica Universita del lAquila

Via Vetoio I Coppito LAquila Italy

3

Dipartimento di Fisica Universita di Roma La Sapienza

Ple AMoro I Roma Italy

ABSTRACT

We discuss the predictability of a conservative system that drives a chaotic system

with p ositive maximum Lyapunov exp onent such as the erratic motion of an asteroid

0

in the gravitational eld of two b o dies of much larger mass We consider the case where

in absence of feedback restricted mo del the driving system is regular and completely

predictable A small feedback of strength still allows a go o d forecasting in the driving

system up to a very long time T where dep ends on the details of the system

p

The most interesting situation happ ens when the Lyapunov exp onent of the total system

is strongly chaotic with practically indep endent of Therefore an exp onential

tot 0

amplication of a small incertitude on the initial conditions in the driving system for any

co exists with very long predictability times The paradox stems from saturation

eects in the evolution for the growth of the incertitude as illustrated in a simple mo del

of coupled maps and in a system of three p oint vortices in a disk

PACS NUMBERS b

It is commonly b elieved that a sensible dep endence on initial condition makes foreca

sting imp ossible even in systems with few degrees of freedom This is the socalled buttery

eect discovered by Lorenz in a numerical simulation of a mo del of convection with three

degrees of freedom Using his words A buttery moving its wings over Brazil might

cause the formation of a tornado in Texas

In general a dynamical system is considered chaotic when there is an exp onential

amplication of an innitesimal p erturbation on the initial conditions with a mean

0

time rate given by the inverse of the maximum Lyapunov exp onent Indeed such a

1

deterministic system is exp ected to b e predictable on times t and to b ehave like

a random system on larger times The purp ose of this letter is to show that there exists

a wide class of dynamical systems where a large value of the Lyapunov exp onent do es

not imply a short predictability time on a physically relevant part of the system In this

resp ect one can sp eak of strong chaos without buttery eect

In particular we discuss the predictability of a conservative system that drives a stron

gly chaotic system with p ositive maximum Lyapunov exp onent In absence of feedback

0

the driving system is regular and completely predictable A small feedback of strength

still allows to predict the future of the driving system up to a very long predictability time

T that diverges with The Lyapunov exp onent of the total system is and so

p tot 0

there is a regime of strong chaos for all values The absence of the buttery eect stems

from saturation eects in the evolution laws for the growth of an incertitude on the driven

system

To b e explicit let us consider a system with evolution given by two sets of equations

d

F h a

dt

d

G b

dt

n m

with R and R The variables are thus driven by a subsystem represented by

the variables with a weak feedback of order A typical physical example is given by an

asteroid moving in the gravitational eld generated by two celestial b o dies of much larger

mass such as Jupiter and Sun Usually the feedback is neglected and one considers

the restricted three b o dy problem ie the variables passively driven However a ner

description should take into account even the inuence of the asteroid on the evolution

of the other two b o dies ie an active driving where This situation app ears in

many other phenomena such as the active advection of a contaminant in a uid a simple

example which will b e discussed in this letter

The main prop erties of the system in absence of feedback are the following

the driver is an indep endent dynamical system that exhibits a regular evolution with

zero Lyapunov exp onent

the driven subsystem is chaotic with p ositive maximum Lyapunov exp onent say

0

In other terms the b ehavior of the driver variables is completely predictable

However as so on as one should consider the total system which is obviously chaotic

with a Lyapunov exp onent that a part small correction of order is given by the

tot

Lyapunov exp onent of the chaotic driven subsystem This means that there is an

0

exp onential amplication of a small incertitude on the knowledge of the initial conditions

even in the driver This is an amazing result since it is natural to exp ect that it is p ossible

to forecast the b ehavior of the driver for very long times as Actually intuition

is correct while the Lyapunov analysis gives completely wrong hints on the predictability

problem at dierence with what commonly b elieved The famous buttery eect of Lorenz

seems not to forbid the p ossibility of predicting the future of a part of the system The

paradox stems from saturation eects in the evolution for the growth of the incertitude

To x notation and denitions let us consider the evolution of the total system

dx

i

n+m

f x where x R

i

dt

0 0

The incertitude on its state is t xt x t where x and x are tra jectories starting

0

from close initial conditions ie jx x j In the limit can b e confused

0 0

with the the tangent vector z whose evolution equations are

df dz

i

J t z where J t

ik

dt dx

j

x(t)

The maximum Lyapunov exp onent is then dened as the exp onential rate of the incertitude

growth

jtj

lim lim lim ln ln jztj

t!1 t!1

!0

t t

0

0

It is worth stressing that the full equations for the evolution of an incertitude are nonlinear

d

2

J t O

dt

so that the two limits in cannot b e interchanged

The predictability of the system is dened in terms of the allowed maximal ignorance

on the state of the system a tolerance parameter which must b e xed according the

max

necessities of the observer The predictability time is thus

0 0

T sup ft such that jt j for t tg

p max

t

If is well approximated by and the predictability time can b e roughly

max

identied with the inverse Lyapunov exp onent since

max

ln T

p

0

and the dep endence on the initial error and on the tolerance parameter is only

0 max

logarithmic and can b e safely ignored for many practical purp oses

( )

Supp ose now to b e interested only on the incertitude in the driver system When

the Lyapunov exp onent of the driver so that it is fully predictable we

( )

have in general T the exp onent dep ending on the particular system while

p

0

the driven system has a Lyapunov exp onent However for any the two

0

subsystems are coupled and the global Lyapunov exp onent is exp ected to b e

O

0

A direct application of would give

( )

T T

p

p

0

( ) ( )

One thus obtains a singular limit lim T T The troubles stem from

p p

!0

the identication b etween incertitude t and tangent vector zt which is not correct

on long time scales

It is convenient to illustrate the problem in a simpler context as the main qualitative

asp ects of equations a and b can b e repro duced considering only the feedback eect

in two coupled maps of the type

L h a

t+1 t t

G b

t+1 t

2

where the time t is an integer variable R h h h is a vector function of the

1 2

1

variable R whose evolution is ruled by a chaotic onedimensional map G and L is

the linear op erator corresp onding to a rotation of an arbitrary angle

cos sin

L

sin cos

When one is left with two indep endent systems one of them regular and of Hamil

tonian type the other fully chaotic

These maps provide a simple maybe the simplest example of a system with two

dierent temp oral regimes

A short times where exp t so that it is correct to ignore the nonlinear term in

0 0

so that z

B long times where one should consider the full nonlinear equation for the incertitude

growth

From the observer p oint of view b oth these regimes might b e interesting according his

particular exigence If one is interested in forecasting the very ne details of the systems

1

the tolerance threshold could b e quite small hence T In general however a

max p

0

system is considered unpredictable when the incertitude is rather large say discrimination

b etween sunrain in meteorology and regime B is the relevant one In that case non

linear eects in cannot b e neglected and in order to give an analytic estimate of the

predictability time we can use a sto chastic mo del of the deterministic equations Indeed

the chaotic feedback on the evolution of the driver system can b e simulated by a random

vector w ie

L w

t+1 t t

( )

0

The incertitude is then given by the dierence of two tra jectories and originated

t

t t

by nearby initial conditions and evolves according the sto chastic map

( ) ( )

L W

t

t

t+1

1

0

where W w w For short times t j ln j one cannot consider the noises w

t t 0 t

t

0

0

and w as uncorrelated so that the incertitude on the driver grows exp onentially under

t

( )

the inuence on the deterministic chaos given by the feedback j j j j exp t

t

t

1

with O For long times t j ln j the random variables are practically

0 0

0

uncorrelated so that their dierence W still acts as a noisy term As a consequence the

t

growth of the incertitude is diusive since the formal solution of is

t

X

( )

t

L W L

0

t

=0

t

and noting that L is a unitary transformation from one can derive the b ound

t

X

( )

12

t W j j

t

=0

P

t

12

where we have used the estimate j W j t given by standard arguments b orrowed

=0

from the central limit theorem In conclusion for our mo del maps a and b the

predictability time on the driver diverges like

2 ( )

T

p

although there is a regime of strong chaos since the total Lyapunov exp onent O

0

do es not vanish with the strength of the feedback

It is imp ortant to stress that the particular p ower of the diusive law in a realistic

mo del can b e dierent from that of a random walk since the deterministic chaos of the

feedback could b e b etter represented by random variables with appropriate correlations

The qualitative b ehavior exhibited by the sto chastic mo del for the incertitude growth

exp onential followed by a p ower law can b e tested in a direct numerical simulation of the

coupled maps a and b where we choose the linear vector function for the feedback

h a

and the logistic map at the Ulam p oint for the driving system

G b

( )

with Lyapunov exp onent ln Fig shows the b ehaviors of the incertitude j j

0

for the driver at starting with an error on the initial condition and no error on the

0 0

( ) ( )

driver At the b eginning b oth j j and j j grows exp onentially However the phase

( )

space available to the variable is nite so that j j is b ounded by a maximum value

1



O It will b e attained at the time t t ln when the incertitude

M M 0

0

( )

on the driver system is j j and so much lower than the threshold At larger

M

 ( )

times t t the incertitude on the driven system remains practically constant and j j

increases with a diusive law of type according to the mechanism describ ed by the

sto chastic map

We have also studied a more realistic mo del of two coupled standard maps in action

angle variables I and

(1) (1) (1) (2)

I I sin

t t t

t+1

(1) (1) (1)

I

t

t+1 t+1

(2) (2) (1) (2)

I I K sin

t t t

t+1

(2) (2) (2)

I

t

t+1 t+1

(2) (2)

where K is a control parameter of order unity such that the system I is

chaotic when We do not discuss in details the results for these coupled maps since

they are qualitatively similar to those obtained for the simplied mo del

We now consider an application to a physical phenomenon the motion of an ensemble

of p oint vortices in a uid It is a classical problem in uid mechanics formally similar to

the planetary motion in gravitational eld Both the systems are Hamiltonian with long

range interactions The main qualitative dierences are that the Hamiltonian for p oint

vortices do es not contain a kinetic term and the motion is conned on the twodimensional

plane The phase space for a collection of N vortices has thus N dimensions related to

the physical co ordinates

Dynamical prop erties of p oint vortex systems have b een studied by several authors

interested in their chaotic motion and connection with two dimensional turbulence see

for a review The Hamiltonian theory for vortex motion inside a b ounded domain was

developed many years ago for several b oundaries We are here interested in the motion

in the unitary disk D for which the Hamiltonian takes the form

N

2 2

X X

r r r r cos

i j ij

i j

2 2

log log r H

i j

i i

2 2

r r cos r r

i j ij

j i

ij i=1

where represent the circulation of the ith vortex of co ordinates x x r cos y

i i i i i i

r sin and The canonical conjugated variable are the scaled co ordinates

i i ij i j

x y and the phase space is thus N times the conguration space D The Hamiltonian

i i i

is invariant under rotations in the conguration space thus the angular momentum

is a second conserved quantity

N

X

2 2 2

x y L

i

i i

i=1

By general results of Hamiltonian mechanics a system of two p oint vortices is always

integrable but we should exp ect chaotic motion for N vortices

In the following we will consider N vortices two of them carrying xed circulation

and representing the driver that without feedback is integrable The third

1 2

vortex of circulation represents the driven system which now makes the total system

3

chaotic In the limit the third vortex b ecomes a passive particle it is passively

transp orted by the ow generated by the two unit vortices and do es not inuence the

integrable motion of the two vortices as for the threeb o dy restricted problem in celestial

mechanics The restricted system is still chaotic but the incertitude is conned to the

passive tracer while the motion of the two vortices is in general quasip erio dic This

limit is one of the simplest example of chaotic advection in twodimensional ow and it

will b e studied in detail in another pap er

For our particular problem of three vortices the Hamiltonian can b e rewritten in the

following standard p erturbation form

2

H H x x H x x x H x

0 1 2 1 1 2 3 2 3

The rst term H describ es the dynamics of the two unit vortices and leads to integrable

0

motion for the term H represents the interaction with the small vortex of circu

1

lation and the last term is due to the interaction of the third vortex with its own image

The O term is thus the p erturbation to the integrable system H and we are reduced

0

to the general framework describ ed ab ove if we identify x x and x The only

1 2 3

dierence is that now the dynamics is not chaotic by itself but chaoticity is induced by

the interaction with the integrable system

We now describ e a typical simulation of error growth in the p oint vortex mo del which

repro duces the eects obtained with the coupled maps mo del We x the value of the

6

coupling constant circulation of the third vortex and the initial condition for

the vortex p ositions are chosen in order to obtain chaotic motion with a global Lyapunov

exp onent The initial incertitude on the co ordinates of the small vortex is

( ) 3

while we supp ose to know the initial p osition of the two big vortices with

( ) 8

a precision of The saturation value for the incertitude is prop ortional to

the disk radius here

M

We let the system evolve according to the Hamiltonian dynamics for quite long

time and we computed at each time the maximum value reached by the incertitude we

used the maximum b ecause in this system incertitude show strong oscillations this is

a memory of the quasip erio dic b ehavior for This represent the worst situation

for making predictions The upp er scatter plot in g shows the time evolution of the

( )

incertitude for the driven system t We can recognize a short t exp onential



growth until the nonlinear eect b ecomes imp ortant At large time t t the

incertitude saturates to its maximum value The lower scatter plot represents the

M

( )

incertitude for the driver system of two vortices t We can easily recognized the two

exp ected limiting b ehavior represented by the two lines For small times the error grows

( ) t

exp onentially t e where is close to the global Lyapunov exp onent

( ) 1 1

For long time the p ower law b ehavior is recovered t t with

In conclusion we must stress that all our results can generalized in a straightforward

way to a weakly chaotic driver with a maximum Lyapunov exp onent In fact

d 0

the driver might b e either conservative or dissipative The imp ortant p oint is that the

dynamics of the driver has a much longer characteristic time than the driven system so that

for an observer interested in the predictability problem the two systems can b e practically

decoupled The Lyapunov analysis although mathematically correct do es not capture the

physically relevant features of the phenomenon and the exp onential dep endence on initial

conditions do es not aect the p ossibility of forecasting the future of the driver on very long

time scale This is still true in systems with many dierent time scales instead of only two

ones as fully developed turbulence where the inverse Lyapunov exp onent is not related

to the predictability time on the large length scale motion

Acknowledgements

We are grateful for nancial supp ort to INFN through Iniziativa Specica FI GB

thanks the Istituto di Cosmogeosica del CNR Torino for hospitality

Figure captions

( )

Fig Growth of the incertitude j j of the driver system in the coupled maps a and

b as a function of time t where the rotation angle the feedback

5

strength and the error on the initial condition of the driven system b

10 ( )

Dashed line exp onential regime t exp t where ln

0 0 0 0

( ) 12

Full line t t

Fig Incertitude growth for the p oint vortex mo del Cross maximum of the incertitude

( ) ( )

t on the third vortex Diamond maximum error on the driving system

( )

of two vortices Dotted line exp onential regime t exp t with

( ) 1 1

Dashed line p ower law regime t with

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