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1 Riddling Transition in Unidirectionally-Coupled Chaotic Systems Sang-Yoon Kim Department of Physics Kangwon National University Korea Synchronization in Coupled Periodic Oscillators Synchronous Pendulum ClocksSynchronously Flashing Fireflies

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2 Chaos and Synchronization Lorenz Attractor [Lorenz, J. Atmos. Sci. 20, 130 (1963).] Coupled Brusselator Model (Chemical Oscillators) [H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983).] z y x Butterfly Effect: Sensitive Dependence on Initial Conditions (small cause large effect) Other Pioneering Works A.S. Pikovsky, Z. Phys. B 50, 149 (1984). V.S. Afraimovich, N.N. Verichev, and M.I. Rabinovich, Radiophys. Quantum Electron. 29, 795 (1986). L.M. Pecora and T.L. Carrol, Phys. Rev. Lett. 64, 821 (1990).

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3 Frequency (kHz) Secret Message Spectrum Chaotic Masking Spectrum Chaotic System + Chaotic System - Secure Communication (Application) Transmission Using Chaotic Masking Transmitter Receiver (Secret Message) Several Types of Chaos Synchronization Different degrees of correlation between the interacting subsystems Identical Subsystems Complete Synchronization [H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983).] Nonidentical Subsystems Generalized Synchronization [N.F. Rulkov et.al., Phys. Rev. E 51, 980 (1995).] Phase Synchronization [M. Rosenblum, A.S. Pikovsky, and J. Kurths, Phys. Rev. Lett 76, 1804 (1996).] Lag Synchronization [M. Rosenblum, A.S. Pikovsky, and J. Kurths, Phys. Rev. Lett 78, 4193 (1997).] [K.M. Cuomo and A.V. Oppenheim, Phys. Rev. Lett. 71, 65 (1993).]

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4 Period-doubling transition to chaos An infinite sequence of period doubling bifurcations ends at a finite accumulation point A = 1D Map (Building Blocks) Chaos Synchronization in Unidirectionally Coupled 1D Maps Unidirectionally Coupled 1D Maps Invariant synchronization line y = x Synchronous orbits on the diagonal Asynchronous orbits off the diagonal

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5 Transverse Stability of The Synchronous Chaotic Attractor Synchronous Chaotic Attractor (SCA) on The Invariant Synchronization Line SCA: Stable against the “Transverse Perturbation” Chaos Synchronization An infinite number of Unstable Periodic Orbits (UPOs) embedded in the SCA and forming its skeleton Characterization of the Macroscopic Phenomena Associated with the Transverse Stability of the SCA in terms of UPOs (Periodic-Orbit Theory)

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6 Transverse Bifurcations of UPOs : Transverse Lyapunov exponent of the SCA (determining local transverse stability) (SCA Transversely stable) Chaos Synchronization (SCA Transversely unstable chaotic saddle) Complete Desynchronization {UPOs} = {Transversely Stable Periodic Saddles (PSs)} + {Transversely Unstable Periodic Repellers (PRs)} “Weight” of {PSs} > ( < ) “Weight” of {PRs} Investigation of transverse stability of the SCA in terms of UPOs Chaos Synchronization Blowout Bifurcation Blowout Bifurcation C

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7 A Transition from Strong to Weak Synchronization Weak Synchronization Strong Synchronization 1st Transverse Bifurcation C Attracted to another distant attractor Dependent on the existence of an Absorbing Area, controlling the global dynamics and acting as a bounded trapping area Folding back of repelled trajectory (Attractor Bubbling) Local Stability Analysis: Complemented by a Study of Global Dynamics (Basin Riddling) 1st Transverse Bifurcation All UPOs embedded in the SCA: transversely stable PSs Strong Synchronization A 1st PS becomes transversely unstable via a local Transverse Bifurcation. Local Bursting Weak Synchronization Fate of Local Bursting?

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8 Bubbling Transition through The 1st Transverse Bifurcation C Strong synchronizationBubblingRiddling Case of Presence of an absorbing area Bubbling Transition Noise and Parameter Mismatching Persistent intermittent bursting (Attractor Bubbling) Transient intermittent bursting Transcritical Contact Bif. Supercritical Period-Doubling Bif.

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9 Riddling Transition through A Transcritical Contact Bifurcation Disappearance of An Absorbing Area through A Transcritical Contact Bifurcation : saddle : repeller

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10 C Strong synchronizationBubblingRiddling Case of Transcritical Contact Bif. Supercritical Period-Doubling Bif. Disappearance of an absorbing area Riddling Transition an absorbing area surrounding the SCA Contact between the SCA and the basin boundary

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11 Riddled Basin After the transcritical contact bifurcation, the basin becomes “riddled” with a dense set of “holes” leading to divergent orbits. The SCA is no longer a topological attractor; it becomes a Milnor attractor in a measure-theoretical sense. As C decreases from C t,l, the measure of the riddled basin decreases.

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12 Characterization of The Riddled Basin Divergence Probability P(d) Take many randomly chosen initial points on the line y=x+d and determine which basin they lie in Measure of the Basin Riddling Superpower-Law Scaling Power-Law Scaling C Power LawSuperpower Law Blow-out Bifurcation Riddling Transition Crossover Region ~~

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13 Uncertainty Exponent Probability P( ) Take two initial conditions within a small square with sides of length 2 inside the basin and determine the final states of the trajectories starting with them. Fine Scaled Riddling of the SCA Superpower-Law Scaling Power-Law Scaling

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14 Phase Diagram for The Chaotic and Periodic Synchronization Hatched Region: Strong Synchronization, Light Gray Region: Bubbling, Dark Gray Region: Riddling Solid or Dashed Lines: First Transverse Bifurcation Lines, Solid Circles: Blow-out Bifurcation C A

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15 First Transverse Bifurcation Riddling transition occurs through a Transcritical Contact Bifurcation [ S.-Y. Kim and W. Lim, Phys. Rev. E 64, (2001). S.-Y. Kim, W. Lim, and Y. Kim, Prog. Theor. Phys. 105, 187 (2001). ] The same kind of riddling transition occurs also with nonzero (0 < 1) in general asymmetric systems [ S.-Y. Kim and W. Lim, Phys. Rev. E 64, (2001).] Blow-out Bifurcation Investigation of The Mechanism for The Loss of Chaos Synchronization in terms of Transverse Bifurcations of UPOs embedded in The SCA (Periodic-Orbit Theory) Chaotic Saddle Weakly-stable SCA Strongly-stable SCA Summary Such riddling transition seems to be a “Universal” one occurring in Asymmetric Systems

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16 Direct Transition to Bubbling or Riddling Asymmetric systems Transcritical bifurcation Subcritical pitchfork or period-doubling bifurcation Contact bifurcation (Riddling) Non-contact bifurcation (Bubbling of hard type) Symmetric systems (Supercritical bifurcations Bubbling transition of soft type) Contact bifurcation (Riddling) Non-contact bifurcation (Bubbling of hard type) [Y.-C. Lai, C. Grebogi, J.A. Yorke, and S.C. Venkataramani, Phys. Rev. Lett. 77, 55 (1996).] [S.-Y. Kim and W. Lim, Phys. Rev. E 63, (2001).]

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17 Transition from Bubbling to Riddling Boundary crisis of an absorbing area Appearance of a new periodic attractor inside the absorbing area BubblingRiddling BubblingRiddling [Y.L. Maistrenko, V.L. Maistrenko, O. Popovych, and E. Mosekilde, Phys. Rev. E 60, 2817 (1999).] [V. Astakhov, A. Shabunin, T. Kapitaniak, and V. Anishchenko, Phys. Rev. Lett. 79, 1014 (1997).]

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18 ( : constraint-breaking parameter) Superpersistent Chaotic Transient Parameter Mismatch Average Lifetime: ( : some constants)

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19 Chaotic Contact Bifurcation Saddle-Node Bifurcation (Boundary Crisis) Transcritical Bifurcation Subcritical Pitchfork Bifurcation x: Strongly unstable dir. y: Weakly unstable dir. Superpersistent Chaotic Transient average life time: Superpersistent Chaotic Transient (Constraint-breaking: ) Superpersistent Chaotic Transient (Symmetry-breaking: ) (x*: fixed point of the 1D map) ( : saddle-node bif. point)

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