Riddling Transition in Unidirectionally-Coupled Chaotic Systems

1. Riddling Transition in Unidirectionally-CoupledChaotic Systems • Sang-Yoon Kim • Department of Physics • Kangwon National University • Korea Synchronization in Coupled Periodic Oscillators Synchronous Pendulum Clocks Synchronously Flashing Fireflies

2. Chaos and Synchronization [Lorenz, J. Atmos. Sci. 20, 130 (1963).] z Butterfly Effect: Sensitive Dependence on Initial Conditions (small cause  large effect) • Lorenz Attractor y x Coupled Brusselator Model (Chemical Oscillators) [H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983).] •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. Secure Communication (Application) Chaotic Masking Spectrum Secret Message Spectrum Frequency (kHz) [K.M. Cuomo and A.V. Oppenheim, Phys. Rev. Lett. 71, 65 (1993).] Transmission Using Chaotic Masking (Secret Message) Chaotic System Chaotic System  + - Transmitter Receiver 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).]

4. Chaos Synchronization in Unidirectionally Coupled 1D Maps  1D Map (Building Blocks) • Period-doubling transition to chaos An infinite sequence of period doubling bifurcations ends at a finite accumulation point A=1.401 155 189 092 506   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 Investigation of transverse stability of the SCA in terms of UPOs {UPOs} = {Transversely Stable Periodic Saddles (PSs)} + {Transversely Unstable Periodic Repellers (PRs)} “Weight” of {PSs} > (<) “Weight” of {PRs}  Chaos Synchronization C Blowout Bifurcation Blowout Bifurcation

7. A Transition from Strong to Weak Synchronization Weak Synchronization Strong Synchronization Weak Synchronization C 1st Transverse Bifurcation 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? Dependent on the existence of an Absorbing Area, controlling the global dynamics and acting as a bounded trapping area Attracted to another distant attractor Folding back of repelled trajectory (Attractor Bubbling) (Basin Riddling) Local Stability Analysis: Complemented by a Study of Global Dynamics

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

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

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

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 Ct,l, the measure of the riddled basin decreases.

12. Characterization of The Riddled Basin Power Law Superpower Law C Blow-out Bifurcation Crossover Region Riddling Transition ~ ~  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

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. 2.0 1.8 A 1.6 1.4 -3.4 -2.6 -0.8 0.0 C 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

15. Summary Investigation of The Mechanism for The Loss of Chaos Synchronization in terms of Transverse Bifurcations of UPOs embedded in The SCA (Periodic-Orbit Theory) First Transverse Bifurcation Strongly-stable SCA Weakly-stable SCA Chaotic Saddle Blow-out Bifurcation  Riddling transition occurs through a Transcritical Contact Bifurcation [S.-Y. Kim and W. Lim, Phys. Rev. E 64, 016211 (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, 016211 (2001).] • Such riddling transition seems to be a “Universal” one occurring in Asymmetric Systems

16. Direct Transition to Bubbling or Riddling (Supercritical bifurcations  Bubbling transition of soft type) • Symmetric systems Subcritical pitchfork or period-doubling bifurcation Contact bifurcation (Riddling) [Y.-C. Lai, C. Grebogi, J.A. Yorke, and S.C. Venkataramani, Phys. Rev. Lett. 77, 55 (1996).] Non-contact bifurcation (Bubbling of hard type) • Asymmetric systems Transcritical bifurcation Contact bifurcation (Riddling) [S.-Y. Kim and W. Lim, Phys. Rev. E 63, 026217 (2001).] Non-contact bifurcation (Bubbling of hard type)

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

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

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