Dynamical Origin for the Occurrence of Asynchronous Hyperchaos and Chaos via Blowout Bifurcations W. Lim1, S.-Y. Kim1, E. Ott2, and B. Hunt2 1 Department of Physics Kangwon National University, 2 University of Maryland, U.S.A.  System Coupled 1D Maps: • : Parameter Tuning the Degree of Asymmetry of Coupling (01) • c: Coupling Parameter  New Coordinate • Invariant Synchronization Line: v=0 Synchronous Orbits Lie on the Invariant Synchronization Line.

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* (=0.2299), =0.  Blow-Out Bifurcation • SCA: Transversely Unstable • Appearance of an Asynchronous Attractor (Its type is determined by the sign of its 2nd Lyapunov exponent.)

Type of Asynchronous Attractors Born via Blow-Out Bifurcations  Second Lyapunov Exponents of the Asynchronous Attractors a=1.97 : =0 : =0.852 : =1 Threshold Value * ( 0.852) s.t. •  < *  Hyperchaotic Attractor (HCA) with 2 > 0 •  > *  Chaotic Attractor (CA) with 2 < 0 (Total Length of All Segments L=108) s=s-s* HCA for  = 0 CA for  = 1 a=1.97 s=-0.0016 a=1.97 s=-0.0016 1 0.6087 2 0.0024 1 0.6157 2 -0.0028

Mechanism for the Transition from Hyperchaos to Chaos  On-Off Intermittent Attractors born via Blow-Out Bifurcations  = 0  = 1 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 Laminar and 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 • Sign of 2 Threshold Value * ( 0.852) s.t.  < * HCA with 2 > 0  > * CA with 2 < 0

Blow-Out Bifurcations in High Dimensional Invertible Systems  System: Coupled Hénon Maps New Coordinate • Type of Asynchronous Attractors Born via Blow-Out Bifurcations (s*=0.787 for b=0.1 and a=1.83) L=108 d *=10-4 d *=10-4 Threshold Value * ( 0.905) s.t.

 System: Coupled Parametrically Forced Pendulums HCA with 2 > 0, for  > * CA with 2 < 0 HCA for  = 0 CA for  = 1 a=1.83, s=-0.0016 a=1.83, s=-0.0016 1 0.4406 2 -0.0024 1 0.4340 2 0.0031  System: Coupled Parametrically Forced Pendulums New Coordinate

• Type of Asynchronous Attractors Born via Blow-Out Bifurcations (s*=0.324 for =1.0, =0.5, and A=0.85) L=107 d *=10-4 d *=10-4 Threshold Value * ( 0.84) s.t. For  < * HCA with 2 > 0, for  > * CA with 2 < 0 HCA for  = 0 CA for  = 1 A=0.85 s =-0.006 A=0.85 s=-0.005 1 0.628 2 0.017 1 0.648 2 -0.008

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