1. 1 Effect of Asymmetry on Blow-Out Bifurcations in Coupled Chaotic Systems  System Coupled 1D Maps: Invariant Synchronization Line: y = x  =0: Symmetrical Coupling Case  0: 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. Kim Department of Physics Kangwon National University

2. 2 One-Band SCA on the Invariant Diagonal Transverse Stability of the Synchronized Chaotic Attractor (SCA) Longitudinal Lyapunov exponent of the SCA Transverse Lyapunov exponent of the SCA 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: a=1.83

3. 3 Type of Asynchronous Attractors Born via Blow-Out Bifurcations Threshold Value  * ( 0.77) s.t.  <  *  Hyperchaotic Attractor (HCA) with > 0  >  *  Chaotic Attractor (CA) with < 0  Second Lyapunov Exponents of the Asynchronous Attractors HCA for  = 0  1 0.471  2 0.015  1 0.478  2 -0.001 CA for  = 1 a=1.83 s=0.187 a=1.83 s=0.187 (Total Length of All Segments L t =5  10 7 )

4. 4 Mechanism for the Transition from Hyperchaos to Chaos d * : Threshold Value for the Laminar State  Decomposition of into the Sum of the Weighted 2nd Lyapunov Exponents of the Laminar and Bursting Components (i=l, b); L i : 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 : “Weighted” 2nd Lyapunov Exponent for the Laminar (Bursting) Component. ’  On-Off Intermittent Attractors born via Blow-Out Bifurcations  = 0  = 1 d < d * : Laminar State (Off State), d  d * : Bursting State (On State)

5. 5 Competition between the Laminar and Bursting Components C l : Independent of  C b : Decrease with Increasing  Threshold Value  * ( 0.77) s.t.  <  *  >  * HCA with > 0 CA with < 0 a=1.83 d * =10 -4 a=1.83 d * =10 -4  Dependence of the Slopes of on  Sign of (s * =0.1895)

6. 6  System: Coupled Hénon Maps Type of Asynchronous Attractors Born via Blow-Out Bifurcations Threshold Value  * ( 0.9) s.t. For  <  * for  >  * HCA with > 0,CA with < 0 (s*=0.1674 for b=0.1 and a=1.8) d * =10 -4 L t =5  10 7 Blow-Out Bifurcations in High Dimensional Invertible Systems HCA for  = 0 CA for  = 1 a=1.8, s=0.165  1 0.398  2 -0.002  1 0.382  2 0.014

7. 7  System: Coupled Parametrically Forced Pendulums Threshold Value  * ( 0.8) s.t. For  <  * for  >  * HCA with > 0, CA with < 0 HCA for  = 0 CA for  = 1  1 0.185  2 0.002  1 0.190  2 -0.002 A=0.3585 S=0.093 A=0.3585 S=0.093 L t =10 6 d * =10 -4 Type of Asynchronous Attractors Born via Blow-Out Bifurcations (s*=0.094 for  =0.2,  =0.5, and A=0.3585)

8. 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  > 0. For  >  *, CA with < 0.  Similar Result: Found in the High-Dimensional Invertible Systems such as Coupled Hénon Maps and Coupled Parametrically Forced Pendulums