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

1. 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.

2. 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

3. 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

4. ’ 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

5. 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  > *

6. 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

7. 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

8. 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