Download - Lyapunov Exponent
1
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
2
Lyapunov Exponent( σ )
•Perturbation : tt xx *
)*(* 1 tAq
t xfx
•Let be the separation after t-th
iteration.
t
●●
●●
●●
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
3
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.