1 / 8

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

Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related