Lyapunov Exponent of The 2D Henon Map

The Lyapunov exponent characterize the exponential divergence of a nearby orbit.

In the 2D Map, attractor with one or two positive Lyapunov exponent is said to be chaotic attractor.

• 2D Henon Map:nnn yAxx

21 1

Eui-Sun Lee

Department of Physics

Kangwon National University

nn xby 1

• Two Lyapunov exponents

The two Lyapunov exponents( ) is allowed along the two eigen-directions in the 2D Henon Map.

21,

,02 DetMTrM .21 bDetM

)ln()ln( 2121

The total summation of Lyapunov exponent is the constant for the given b .

).ln(b

Gram –Schmidt Reorthonormalization(GSR) Process

• Reorthonormalization algorithm in the 2D Map

1 step:

2 step:

Set the two initial vector , where

Measure the largest Lyapunov exponent

3 step: Reorthonormalization process.

}.1,0{)0(},0,1{)0( 21 ww

• The Introduction to GSR Method.

There are two problem to calculate the Lyapunov exponents.

1.The two evolved vector by the linearized map tends to align with each other to the rapid growth direction.

2.The limited storage of computer can’t store up the exponential growth of the evolved vector.

.||)0()0(||)1( 11 wMd

,)1(

)0()0()1(

1

11

d

wMw

.)1(

)1()1(

2

12

d

uw

).0()]0()0()0([)0()0()1( 11222 wwwMwMu

||,)1(||)1( 22 ud

)}0(),0({ 21 ww

• Reorthonormalizated sets: )}1(),1({ 21 ww

• Lyapunov exponent: , where

t

i

j

tj id

t 1

)](ln[1

lim .2,1j

Orthogonalization process .

Normalization process .

)}2(),2({, 21 ww

.)},3(),3({, 21 ww

Period Doubling Route to Chaos

• Bifurcation and Lyapunov exponent diagram

Transition to chaos occurs when the Lyapunov exponent becomes positive.

In the case of the 2D Henon Map having a constant Jacobian , the Lyapunov exponent spectra exhibits symmetry about the

b=0.3

Bifurcation diagram Lyapunov exponent

].ln[ b