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Effect of Asymmetry on Blow-Out Bifurcations in Coupled Chaotic Systems. W. Lim and S.-Y. Kim Department of Physics Kangwon National University.  System Coupled 1D Maps:. • : Parameter Tuning the Degree of Asymmetry of Coupling. =0: Symmetrical Coupling Case

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effect of asymmetry on blow out bifurcations in coupled chaotic systems
Effect of Asymmetry on Blow-Out Bifurcations in Coupled Chaotic Systems

W. Lim and S.-Y. Kim

Department of Physics

Kangwon National University

 System

Coupled 1D Maps:

• : Parameter Tuning the Degree of Asymmetry of Coupling

=0: Symmetrical Coupling Case

0: Asymmetrical Coupling Case

(=1: Unidirectional Coupling Case)

• c: Coupling Parameter

• Invariant Synchronization Line: y = x

Synchronous Orbits Lie on the Invariant Diagonal.

transverse stability of the synchronized chaotic attractor sca
Transverse Stability of the Synchronized Chaotic Attractor (SCA)

• Longitudinal Lyapunov exponent of the SCA

• Transverse Lyapunov exponent of the SCA

Scaled Coupling Parameter:

One-Band SCA on the Invariant Diagonal

Transverse Lyapunov exponent

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.)

a=1.83

type of asynchronous attractors born via blow out bifurcations

1 0.471

2 0.015

1 0.478

2 -0.001

Type of Asynchronous Attractors Born via Blow-Out Bifurcations

 Second Lyapunov Exponents of the Asynchronous Attractors

a=1.83

Threshold Value * ( 0.77) s.t.

•  < *  Hyperchaotic Attractor (HCA)

with <2> > 0

•  > *  Chaotic Attractor (CA)

with <2> < 0

(Total Length of All Segments Lt=5107)

CA for  = 1

HCA for  = 0

a=1.83

s=0.187

a=1.83

s=0.187

mechanism for the transition from hyperchaos to chaos

Mechanism for the Transition from Hyperchaos to Chaos

 On-Off Intermittent Attractors born via Blow-Out Bifurcations

 = 1

 = 0

d*: Threshold Value for

the Laminar State

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

  • Decomposition of <2> into the Sum of the Weighted 2nd Lyapunov Exponents of the

Laminarand Bursting Components

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

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

Fraction of the Time Spent in the i State

2nd Lyapunov Exponent of i State

competition between the laminar and bursting components
Competition between the Laminar and Bursting Components

a=1.83

d*=10-4

a=1.83

d*=10-4

 Dependence of the Slopes of on 

(s*=0.1895)

Cl: Independent of 

Cb: Decrease with Increasing 

• Sign of <2>

Threshold Value * ( 0.77) s.t.

HCA with <2> > 0

 < *

CA with <2> < 0

 > *

blow out bifurcations in high dimensional invertible systems

1 0.382

2 0.014

1 0.398

2 -0.002

Blow-Out Bifurcations in High Dimensional Invertible Systems

 System: Coupled Hénon Maps

• Type of Asynchronous Attractors Born via Blow-Out Bifurcations

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

d*=10-4

d*=10-4

Lt=5107

Threshold Value * ( 0.9) s.t.

For  < *

HCA with <2> > 0,

CA with <2> < 0

for  > *

HCA for  = 0

CA for  = 1

a=1.8, s=0.165

a=1.8, s=0.165

slide7

1 0.185

2 0.002

1 0.190

2 -0.002

• Type of Asynchronous Attractors Born via Blow-Out Bifurcations

 System: Coupled Parametrically Forced Pendulums

(s*=0.094for=0.2, =0.5, and A=0.3585)

Lt=106

d*=10-4

d*=10-4

Threshold Value * ( 0.8) s.t.

CA with <2> < 0

HCA with <2> > 0,

for  > *

For  < *

HCA for  = 0

CA for  = 1

A=0.3585

S=0.093

A=0.3585

S=0.093

summary
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.

  • Similar Result: Found in the High-Dimensional Invertible Systems such as

Coupled Hénon Maps and Coupled Parametrically Forced Pendulums