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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 (01) • c: Coupling Parameter New Coordinate • Invariant Synchronization Line: v=0 Synchronous Orbits Lie on the Invariant Synchronization Line.
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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.)
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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 * ( ) 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= a=1.97 s=
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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
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Competition between the Laminar and Bursting Components
• Sign of 2 Threshold Value * ( ) s.t. < * HCA with 2 > 0 > * CA with 2 < 0
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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 * ( ) s.t.
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System: Coupled Parametrically Forced Pendulums
HCA with 2 > 0, for > * CA with 2 < 0 HCA for = 0 CA for = 1 a=1.83, s= a=1.83, s= System: Coupled Parametrically Forced Pendulums New Coordinate
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• 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 * ( ) 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
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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. Similar Result: Found in the High-Dimensional Invertible Systems such as Coupled Hénon Maps and Coupled Parametrically Forced Pendulums
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