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Mechanism for Occurrence of Asynchronous Hyperchaos and Chaos via Blowout Bifurcations

Mechanism for Occurrence of Asynchronous Hyperchaos and Chaos via Blowout Bifurcations. Dynamical Origin for the Occurrence of Asynchronous Hyperchaos and Chaos via Blowout Bifurcations. Sang-Yoon Kim Department of Physics Kangwon National University.

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Mechanism for Occurrence of Asynchronous Hyperchaos and Chaos via Blowout Bifurcations

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  1. Mechanism for Occurrence of Asynchronous Hyperchaos and Chaos via Blowout Bifurcations Dynamical Origin for the Occurrence of Asynchronous Hyperchaos and Chaos via Blowout Bifurcations Sang-Yoon Kim Department of Physics Kangwon National University  Fully Synchronized Attractor for the Case of Strong Coupling Breakup of the Chaos Synchronization via a Blowout Bifurcation Asynchronous Chaotic Attractor with One Positive Lyapunov Exponent Asynchronous Hyperchaotic Attractor with Two Positive Lyapunov Exponents

  2. Two Coupled Logistic Maps (Representative Model)  N Globally Coupled 1D Maps  Reduced Map Governing the Dynamics of a Two-Cluster State Two-Cluster State Reduced 2D Map  Globally Coupled Maps with Different Coupling Weight p (N2/N): “coupling weight factor” corresponding to the fraction of the total population in the 2nd cluster =0  Symmetric Coupling Case  Occurrence of Asynchronous Hyperchaos =1  Unidirectional Coupling Case  Occurrence of Asynchronous Chaos Investigation of the Consequence of the Blowout Bifurcation by varying  from 0 to 1.

  3. Transverse Stability of the Synchronous Chaotic Attractor (SCA) • Longitudinal Lyapunov Exponent of the SCA • Transverse Lyapunov Exponent of the SCA a=1.97, s=0.23  One-Band SCA on the Invariant Diagonal Transverse Lyapunov exponent For s>s* (=0.2299), <0   SCA on the Diagonal Occurrence of the Blowout Bifurcation for s=s*  • SCA: Transversely Unstable (>0) for s<s* • Appearance of a New Asynchronous Attractor a=1.97

  4. Type of Asynchronous Attractors Born via a Blowout Bifurcation New Coordinates For the accuracy of numerical calculations, we introduce new coordinates: SCA on the invariant v=0 line Transverse Lyapunov exponent of the SCA a=1.97, s=0.23 a=1.97 • Appearance of an Asynchronous Attractor through a Blowout Bifurcation of the SCA • The Type of an Asynchronous Attractor is Determined by the Sign of its 2nd Lyapunov Exponent2 (2 > 0  Hyperchaos, 2 < 0  Chaos) [ In the system of u and v, we can follow a trajectory until its length L becomes sufficiently long (e.g. L=108) for the calculation of the Lyapunov exponents of an asynchronous attractor.]

  5. Computation of the Lyapunov Exponents 1 and 2 for a Trajectory Segment with Length L Evolution of a Set of Two Orthonormal Tangent Vectors under the Linearized Map Mn [DT(zn), zn (un,vn)]. • Reorthonormalization by the Gram-Schmidt Reorthonormalization Method (Direction of the 1st Vector: Unchanged) ( Has Only the Component Orthogonal to ) • 1st and 2nd Lyapunov Exponents 1 and 2

  6. Second Lyapunov Exponent of the Asynchronous Attractor (: =0, : =0.852, : =1) a=1.97, L=108  Threshold Value * ( ~ 0.852) s.t. •  < *  Asynchronous Hyperchaotic Attractor (HCA) with 2 > 0 • > *  Asynchronous Chaotic Attractor (CA) with 2 < 0 (dashed line: transverse Lyapunov exponent  of the SCA) HCA for  = 0 CA for  = 1 a = 1.97 s = -0.0016 1 = 0.6087 2 = 0.0024 a = 1.97 s = -0.0016 1 = 0.6157 2 = -0.0028

  7. Mechanism for the Occurrence of Asynchronous Hyperchaos and Chaos ’  Intermittent Asynchronous Attractor Born via a Blowout Bifurcation d = |v|: Transverse Variable d*: Threshold Value s.t. d < d*: Laminar Component (Off State), d > d*: Bursting Component (On State). d (t) We numerically follow a trajectory segment with large length L (=108), and calculate its 2nd Lyapunov exponent. • Decomposition of the 2nd Lyapunov Exponent 2of the Asynchronous Attractor : Weighted 2nd Lyapunov Exponent for the Laminar (Bursting) Component Fraction of the Time Spent in the i Component (Li: Time Spent in the i Component) 2nd Lyapunov Exponent of the i Component (primed summation is performed in each i component)

  8. Competition between the Laminar and Bursting Components Laminar Component (: =0, : =0.852, : =1) a=1.97, d*=10-5 Bursting Component Sign of 2 : DeterminedviatheCompetitionoftheLaminarandBurstingComponents Threshold Value * (~ 0.852) s.t. Asynchronous Hyperchaotic Attractor with 2 > 0   < *  > * Asynchronous Chaotic Attractor with 2 < 0 

  9.  Effect of the Threshold Value d* on (: d*=10-6, : d*=10-8, : d*=10-10) a=1.97 •: Dependent on d* As d* Decreases, a Fraction of the Old Laminar Component is Transferred to the New Bursting Component: In the limit d*0, •2 Depends Only on the Difference Between the Strength of the Laminar and Bursting Components.  The Conclusion as to the Type of Asynchronous Attractors is Independent of d*.

  10. Blowout Bifurcations in High Dimensional Invertible Systems  System: Coupled Hénon Maps New Coordinates: • Type of Asynchronous Attractors Born via Blowout Bifurcations (s*=0.787for b=0.1 and a=1.83) L=108 d*=10-4 d*=10-4 (: =0, : =0.905, : =1) Threshold Value * ( 0.905) s.t. For  < *  HCA with 2 > 0, for  > *  CA with 2 < 0.

  11. 1 0.4340 2 0.0031 1 0.4406 2 -0.0024 HCA for  = 0 CA for  = 1 a=1.83, s=-0.0016 a=1.83, s=-0.0016  System: Coupled Parametrically Forced Pendulums New Coordinates:

  12. 1 0.628 2 0.017 1 0.648 2 -0.008 • Type of Asynchronous Attractors Born via Blowout Bifurcations (s*=0.324for=1.0, =0.5, and A=0.85) L=107 d*=10-4 d*=10-4 (: =0, : =0.84, : =1) 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

  13. Summary • Mechanism for the Occurrence of the Hyperchaos and Chaos via Blowout Bifurcations Sign of the 2nd Lyapunov Exponent of the Asynchronous Attractor Born via a Blowout Bifurcation of the SCA: Determined via the Competition of the Laminar and Bursting Components  Occurrence of the Hyperchaos  Occurrence of the Chaos • Similar Results: Found in High-Dimensional Invertible Period-Doubling Systems such as Coupled Hénon Maps and Coupled Parametrically Forced Pendula

  14. 2q q q q q Effect of Asynchronous UPOs on the Bursting Component  Change in the Number of Asynchronous UPOs with respect tos (from the first transverse bifurcation point st to the blow-out bifurcation point s*) •Symmetric Coupling Case (=0) (Period q=11) • Transverse PFB of a Synchronous Saddle • Asynchronous PDB

  15. q q 2q q q q •Unidirectional Coupling Case (=1) (Period q=11) • Asynchronous SNB • Transverse TB • Asynchronous PDB

  16. q q 2q 2q q q q q • Change in the Number of Asynchronous UPOs at the Blow-Out Bifurcation Point s* (=0.190) with respect to (Period q=11, Ns: No. of Saddles, Nr: No. of Repellers) • SNB • Reverse SNB • PDB • Reverse PDB

  17.  Transition from Chaos to Hyperchaos a=1.83 s=0.155 =1 1 0.478 2 0.018 For s = s* ( 0.163), a Transition from Chaos to Hyperchaos Occurs.

  18. Characterization of the On-Off Intermittent Attractors Born via Blow-Out Bifurcations d: Transverse Variable (Denoting the Deviation from the Diagonal) d < d*: Laminar State (Off State) dd*: Bursting State (On State) p=p*: Blow-Out Bifurcation Point • Distribution of the Laminar Length: • Scaling of the Average Laminar Length: • Scaling of the Average Bursting Amplitude:

  19. Phase Diagrams in Coupled 1D Maps  System: Coupled 1D Maps:  Dissipative Coupling Case with g(x, y) = f(y) – f(x) • Periodic Synchronization Symmetric Coupling (=0) Unidirectional Coupling (=1) Horizontal Lines: Longitudinal Bifurcations  Synchronous Period-Doubling Bifurcations, Nonhorizontal Solid and Dashed Lines: Transverse Bifurcations (Solid Lines: Period-Doubling Bifurcations. Dashed Lines for =0 and 1: Pitchfork and Transcritical Bifurcations, Respectively.)

  20. • Chaotic Synchronization Unidirectional Coupling (=1) Symmetric Coupling (=0) Hatched Region: Strong Synchronization, Light Gray Region: Bubbling, Dark Gray Region: Riddling Solid or Dashed Lines: First Transverse Bifurcation Lines (Solid Lines: Period-Doubling Bifurcations. Dashed Lines for =0 and 1: Pitchfork and Transcritical Bifurcations, Respectively.) Solid Circles: Blow-Out Bifurcation

  21.  Inertial Coupling Case with g(x, y) = y – x • Periodic Synchronization Unidirectional Coupling (=1) Symmetric Coupling (=0) Horizontal Lines: Longitudinal Bifurcations  Synchronous Period-Doubling Bifurcations, Nonhorizontal Solid and Dashed Lines: Transverse Bifurcations (Solid Lines: Period-Doubling Bifurcations. Dashed Lines for =0 and 1: Pitchfork and Transcritical Bifurcations, Respectively.)

  22. • Chaotic Synchronization Symmetric Coupling (=0) Unidirectional Coupling (=1) Hatched Region: Strong Synchronization, Light Gray Region: Bubbling, Dark Gray Region: Riddling Solid or Dashed Lines: First Transverse Bifurcation Lines (Solid Lines: Period-Doubling Bifurcations. Dashed Lines for =0 and 1: Pitchfork and Transcritical Bifurcations, Respectively.) Solid Circles: Blow-Out Bifurcation

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