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Lyapunov Exponent

Eui-Sun Lee

Department of Physics

Kangwon National University

Measure of The degree of Sensitivity on The Initial Conditions.

Chaotic Orbit given Initial Condition Periodic Orbit given Initial Condition

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Lyapunov Exponent( σ )

•Perturbation : tt xx *

)*(* 1 tAq

t xfx

•Let be the separation after t-th

iteration.

t

●●

●●

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210

If ,0t

t e then is called the Lyapunov exponent .

|;]/ln[|1

0 tt

.*)()*( 0 xfxf qA

qAt

|;]*)()*(

ln[|1

lim0

0

00

xfxf

t

qA

qA

|;)('|ln1 1

0

*

t

jjxf

t |;)('|ln

1 *1

0j

t

j

xft

*x

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Determination of Sensitivity On the initial condition

;0|)('|ln1

lim1

0

*

t

jJ

txf

t

q

ln

• > 0 : Exponential divergence

• < 0 : Exponential convergence

, Chaotic Attractor.

, Regular Periodic attractor.

Sensitive on the initial condition

Non-sensitive on the initial condition

For the case of Period-q Orbit, the Stability Multiplier is

Hence, The Lyapunove Exponent Becomes

.*)(1

0

q

tA xf

for the Period-q Orbit.