1

Effect of Asymmetry on Blow-Out Bifurcations in Coupled Chaotic Systems

System Coupled 1D Maps:

,

,1:

1

1

tttt

tttt

yfxfcyfy

xfyfcxfxT

• Invariant Synchronization Line: y = x

.1 2axxf

=0: Symmetrical Coupling Case0: Asymmetrical Coupling Case (=1: Unidirectional Coupling Case)

• : Parameter Tuning the Degree of Asymmetry of Coupling

• c: Coupling Parameter

Synchronous Orbits Lie on the Invariant Diagonal.

W. Lim and S.-Y. KimDepartment of PhysicsKangwon National University

2

83.1a

One-Band SCA on the Invariant Diagonal

Transverse Stability of the Synchronized Chaotic Attractor (SCA)

• Longitudinal Lyapunov exponent of the SCA

N

tt

Nax

N 1|| |2|ln

1lim

• Transverse Lyapunov exponent of the SCA

|21|ln|| s

For s=s* (=0.1895), =0.

Blow-Out Bifurcation

• SCA: Transversely Unstable• Appearance of an Asynchronous Attractor (Its type is determined by the sign of its 2nd Lyapunov exponent.)

Transverse Lyapunov exponent

Scaled Coupling Parameter: cs )2/1(

a=1.83

3

Type of Asynchronous Attractors Born via Blow-Out Bifurcations

Threshold Value * ( 0.77) s.t.• < * Hyperchaotic Attractor (HCA) with <2> > 0

• > * Chaotic Attractor (CA) with <2> < 0

Second Lyapunov Exponents of the Asynchronous Attractors

~

HCA for = 0

1 0.4712 0.015

~~

1 0.4782 -0.001

~~

CA for = 1

a=1.83

a=1.83s=0.187

a=1.83s=0.187

(Total Length of All Segments Lt=5107)

4

Mechanism for the Transition from Hyperchaos to Chaos

d *: Threshold Value for the Laminar State

Decomposition of <2> into the Sum of the Weighted 2nd Lyapunov Exponents of the Laminar and Bursting Components

|| 22222 lbbl )(

2bl

(i=l, b); Li: Time Spent in the i State for the Segment with Length L ii

i22

)0( 2 l

:L

Li

i Fraction of the Time Spent in the i State

2nd Lyapunov Exponent of i State

: “Weighted” 2nd Lyapunov Exponent for the Laminar (Bursting) Component.

:1

state

)2(2

in

nii r

L ’

On-Off Intermittent Attractors born via Blow-Out Bifurcations

= 0 = 1

d < d *: Laminar State (Off State), d d *: Bursting State (On State)

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Competition between the Laminar and Bursting Components

Cl: Independent of Cb: Decrease with Increasing

Threshold Value * ( 0.77) s.t. :0|)|~(~|| 222 lblb CC

|)|(|| 22 lblb CC < *

> *

HCA with <2> > 0

CA with <2> < 0

~

a=1.83d *=10-4

a=1.83d *=10-4

Dependence of the Slopes of on )(2

bl

• Sign of <2> |]|[ 22 lb

~

*)()(2 ; ssssC blbl

|)|(|| 22 lblb CC

(s*=0.1895)

6

System: Coupled Hénon Maps

,)],()([)(

,)],()([)1()()2()2(

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

1

)1()1(1

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

ttttttt

ttttttt

bxyxfxfcyxfx

bxyxfxfcyxfx

.1 2axxf

• Type of Asynchronous Attractors Born via Blow-Out Bifurcations

|| 222 lb

Threshold Value * ( 0.9) s.t. :0||~ 222 lb

For < * for > *HCA with <2> > 0, CA with <2> < 0

~ ~

(s*=0.1674 for b=0.1 and a=1.8)

d *=10-4 d *=10-4Lt=5107

2/|)||(| )1()2()1()2( yyxxd

Blow-Out Bifurcations in High Dimensional Invertible Systems

HCA for = 0 CA for = 1

a=1.8, s=0.165 a=1.8, s=0.165

1 0.3982 -0.002

~~

1 0.3822 0.014

~~

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System: Coupled Parametrically Forced Pendulums

),(),,(),(

),()1(),,(),()1(

212222122

121111211

yyctyxfyxxcyx

yyctyxfyxxcyx

.2sin)2cos(22),,( 2 xtAxtxxf cs )2/1(

Threshold Value * ( 0.8) s.t. :0||~ 222 lb

For < * for > *HCA with <2> > 0, CA with <2> < 0

~ ~

HCA for = 0 CA for = 1

1 0.1852 0.002

~~

1 0.1902 -0.002

~~

A=0.3585S=0.093

A=0.3585S=0.093

Lt=106 d *=10-4 d *=10-4

|| 222 lb

• Type of Asynchronous Attractors Born via Blow-Out Bifurcations(s*=0.094 for =0.2, =0.5, and A=0.3585)

2/|)||(| 1212 yyxxd

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Summary

Type of Intermittent Attractors Born via Blow-Out Bifurcations  (investigated in coupled 1D maps by varying the asymmetry parameter )

Determined through Competition between the Laminar and Bursting Components: 

• Laminar Component : Independent of • Bursting Component : Dependent on Due to the Different Distribution of Asynchronous Unstable Periodic Orbits

With Increasing , Decreases Due to the Decrease in .

Threshold Value * s.t.  For < *,   HCA with <2> > 0. For > *,   CA with <2> < 0.

:0||~ 222 lb

|| 22lb

|| 22lb

Similar Result: Found in the High-Dimensional Invertible Systems such as Coupled Hénon Maps and Coupled Parametrically Forced Pendulums

b2 b

2

b2

l2

|| 222 lb

~