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

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).

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).]

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

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)

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

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?

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.

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

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

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.

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 ~~

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

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

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

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).]

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).]

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

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)