Synchronization in Coupled Chaotic Systems

1. Synchronization in Coupled Chaotic 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, “Stability Theory of Synchronized Motion in Coupled-Oscillator Systems,” Prog. Theor. Phys. 69, 32 (1983)

3. Secure Communication (Application) Chaotic Masking Spectrum Transmission Using Chaotic Masking Secret Message Spectrum (Secret Message) Frequency (kHz) Chaotic System Chaotic System  + - Transmitter Receiver Several Types of Chaos Synchronization Different degrees of correlation between the interacting subsystems  Identical Subsystems  Complete Synchronization  Nonidentical Subsystems  Generalized Synchronization Phase Synchronization Lag Synchronization   

4. An infinite sequence of period doubling bifurcations ends at a finite accumulation point Complete Synchronization in Coupled Chaotic 1D Maps  1D Map (Building Blocks) (x: seasonly breeding inset population) Iterates: (trajectory)  Attractor Period-Doubling Transition to Chaos When exceed , a chaotic attractor with positive Lyapunov exponent s appears.  

5. Coupled 1D Maps Coupling function  C: coupling parameter Asymmetry parameter    = 0: symmetric coupling  exchange symmetry  = 1: unidirectional coupling Invariant synchronization line y = x Synchronous orbits on the diagonal Asynchronous orbits off the diagonal 

6. 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)

7. 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}  Weak Synchronization Strong Synchronization Weak Synchronization C Blowout Bifurcation First Transverse Bifurcation First Transverse Bifurcation Blowout Bifurcation

8. Strong Synchronization e.g. Unidirectionally and Dissipatively Coupled Case with  = 1 and g(x, y) = y2-x2 Strong synchronization for A = 1.82 and Ct,l (= -2.789 …) < C < Ct,r (= -0.850 …)  All UPOs embedded in the SCA: Transversely stable SCA: Asymptotically stable (Lyapunov stable + Attraction in the usual topological sense) Attraction without bursting for all t

9. Global Effect of The First Transverse Bifurcation Transverse Bifurcation through which a first periodic saddle becomes transversely unstable Local Bursting  Lyapunov unstable (Loss of Asymptotic Stability) Fate of A Locally Repelled Trajectory? 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 Local Stability Analysis: Complemented by a Study of Global Dynamics

10. BubblingTransitionthroughThe1stTransverseBifurcation 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)

11. RiddlingTransitionthroughThe1stTransverseBifurcation 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

12. 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.

13. Characterization of The Riddled Basin  Divergence Exponent Divergence probability P(d) ~ d (Take many randomly chosen initial conditions on the line y=x+d and determine which basin they lie in.)  Measure of the Basin Riddling  Uncertainty Exponent Uncertainty 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

14. Direct Transition to Bubbling or Riddling (Supercritical bifurcations  Bubbling transition of soft type) • Symmetric systems Subcritical pitchfork or period-doubling bifurcation Contact bifurcation (Riddling) Non-contact bifurcation (Bubbling of hard type) • Asymmetric systems Transcritical bifurcation Contact bifurcation (Riddling) Non-contact bifurcation (Bubbling of hard type)

15. Transition from Bubbling to Riddling • Boundary crisis of an absorbing area Bubbling Riddling • Appearance of a new periodic attractor inside the absorbing area Bubbling Riddling

16. Basin Riddling through A Dynamic Stabilization Symmetrically and dissipatively coupled case with =0 and Riddling Bubbling

17. Global Effect of Blow-out Bifurcations ~ ~ - - Riddling Strong synchronization Bubbling C First Transverse Bif. First Transverse Bif. Blow-out Bif. Blow-out Bif. Successive Transverse Bifurcations: Periodic Saddles (PSs)  Periodic Repellers (PRs) (transversely stable) (transversely unstable) Weight of {PRs} > Weight of {PSs}  SCA  Transversely Unstable Chaotic Saddle  Complete Desynchronization  For C < Cb,l, absence of an absorbing area  Subcritical blow-out bifurcation Abrupt collapse of the synchronized chaotic state • For C > Cb,r, presence of an absorbing area  Supercritical blow-out bifurcation Appearance of an asynchronous chaotic attractor covering the whole absorbing area and exhibiting the On-Off Intermittency

18. Symmetry-Conservingand-BreakingBlow-out Bifurcations Symmetrically and linearly coupled case with =0 and Depending on the shape of a minimal invariant absorbing area, symmetry may or may not be conserved. Symmetry-Breaking Blow-out Bifurcation Symmetry-Conserving Blow-out Bifurcation

19. Type of Asynchronous Attractors Born via Blow-out Bif. {Asynchronous UPOs inside an absorbing area}={Asynchronous PSs with one unstable direction} +{Asynchronous PRs with two unstable directions} Weight of {PRs} > Weight of {PSs} Weight of {PRs} < Weight of {PSs} Hyperchaotic attractor for =0 Chaotic attractor for =1 Numbers of the period-11 saddles (Ns) and repellers (Nr): Nr > Ns for  < 0.8 Nr < Ns for  > 0.9

20. Phase Diagram for The Chaos Synchronization Dissipatively coupled case with 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 Circles: Blow-out Bifurcation

21. 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 Their Macroscopic Effects depend on The Existence of The Absorbing Area.  Subcritical case  Abrupt Collapse of A Synchronous Chaotic State  Supercritical case  Appearance of An Asynchronous Chaotic Attractor, Exhibiting The On-Off Intermittency.  Attractor Bubbling  Basin Riddling References [1] S.-Y. Kim and W. Lim, Phys. Rev. E 63, 026217 (2001). [2] S.-Y. Kim, W. Lim, and Y. Kim, Prog. Theor. Phys. 105, 187-196 (2001). [3] S.-Y. Kim and W. Lim, Phys. Rev. E 64, 016211 (2001).

22. 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 Their Macroscopic Effects depend on The Existence of The Absorbing Area.  Subcritical case  Abrupt Collapse of A Synchronous Chaotic State  Supercritical case  Appearance of An Asynchronous Chaotic Attractor. The type (Symmetric or Asymmetric, Chaotic or Hyperchaotic) of which is determined by an absorbing area.  Attractor Bubbling  Basin Riddling References [1] S.-Y. Kim and W. Lim, Phys. Rev. E 63, 026217 (2001). [2] S.-Y. Kim, W. Lim, and Y. Kim, Prog. Theor. Phys. 105, 187-196 (2001). [3] S.-Y. Kim and W. Lim, Phys. Rev. E 64, 016211 (2001).

23. Universality for The Chaos Synchronization  Mechanisms for The Loss of Chaos Synchronization in Coupled 1D Maps  Are these mechanisms still valid for the real systems such as the coupled Hénon maps and coupled oscillators? I think that those mechanisms are Universal ones, independently of the details of coupled systems, based on our preliminary results.  Universality for The Periodic Synchronization (well understood) The coupled 1D maps and coupled oscillators have the phase diagrams of the same structure and they exhibit the same scaling behavior on their critical set. I believe that there may exist some kind of Universality for both the Chaotic and Periodic Synchronization in Coupled Dynamical Systems. I suggest the Experimentalists to confirm this kind of universality in real experiment such as the electronic-circuit experiment.

24. Phase Diagram for The Chaos Synchronization Linearly coupled case with 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 Circles: Blow-out Bifurcation OpenCircles:Bdry.CrisisofAnAbsorbingArea,OpenSquares:Bdry.CrisisofAnAsyn.ChaoticAttractor

25. Destruction of Hyperchaotic Attractors through The Dynamic Stabilization When a dynamic stabilization occurs before the blow-out bifurcation, a transition from bubbling to riddling takes place. However, a sudden destruction of a hyperchaotic attractor occurs when such a dynamic stabilization occurs after a blow-out bifurcation.

26. Phase Diagram for Destruction of Hyperchaotic Attractors

27. Phase Diagram for The Periodic Synchronization Dissipatively coupled case with Symmetric coupling (=0) Unidirectional coupling (=1)

28. Phase Diagram for The Periodic Synchronization Linearly coupled case with Symmetric coupling (=0) Unidirectional coupling (=1)

29. Effect of Parameter Mismatch and Noise for The Bubbling Case : Mismatching parameter : Noise strength Parameter mismatch or noise  The SCA is broken up, and then it exhibits a persistent intermittent bursting.  Attractor bubbling The maximum bursting amplitude increases when passing C=Ct,r. |y-x|max |y-x|max Ct,r Ct,r

30. Abrupt Change of The Maximum Bursting Amplitude ~ - The maximum bursting amplitude increases abruptly through the interior crisis of the absorbing area for C-0.8437 Small absorbing area before the crisis Large absorbing area after the crisis Abruptincreaseofthemaximumburstingamplitudeisincontrasttothecaseofsymmetriccoupling. Symmetric coupling (=0) Unidirectional coupling (=1) |y-x|max |y-x|max

31. Effect of Parameter Mismatch and Noise for The Riddling Case Parameter mismatch or noise SCA with the riddled basin Chaotic transient

32. Characterization of The Chaotic Transients ~ - : Average life-time of the chaotic transient Ct,l C -2.84 C Algebraic scaling Exponential scaling (long lived chaotic transient) Crossover