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

2. 2 Chaos and Synchronization  Lorenz Attractor [ Lorenz, J. Atmos. Sci. 20, 130 (1963)]  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) z y x Butterfly Effect: Sensitive Dependence on Initial Conditions (small cause  large effect)

3. 3 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  Nonidentical Subsystems  Generalized Synchronization Phase Synchronization Lag Synchronization    Secret Message Spectrum Chaotic Masking Spectrum Frequency (kHz)

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

5. 5 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  Coupled 1D Maps   

6. 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. 7 Weak Synchronization Weak Synchronization Strong Synchronization 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)} Blowout Bifurcation First Transverse Bifurcation First Transverse Bifurcation Blowout Bifurcation “Weight” of {PSs} > ( < ) “Weight” of {PRs}  C Investigation of transverse stability of the SCA in terms of UPOs

8. 8 Strong Synchronization 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 e.g. Unidirectionally and Dissipatively Coupled Case with  = 1 and g(x, y) = y 2  x 2 Strong synchronization for A = 1.82 and C t,l (=  2.789 …) < C < C t,r (=  0.850 …) 

9. 9 Global Effect of The First Transverse Bifurcation Fate of A Locally Repelled Trajectory? Attracted to another distant attractor Transverse Bifurcation through which a first periodic saddle becomes transversely unstable Local Bursting  Lyapunov unstable (Loss of Asymptotic Stability) Dependent on the existence of an Absorbing Area, controlling the global dynamics and acting as a bounded trapping area Folding back of repelled trajectory Local Stability Analysis: Complemented by a Study of Global Dynamics

10. 10 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.

11. 11 C Strong synchronizationBubblingRiddling  Case of Transcritical Contact Bif. Supercritical Period-Doubling Bif. Riddling Transition through The 1st Transverse Bifurcation Disappearance of an absorbing area  Riddling Transition an absorbing area surrounding the SCA Contact between the SCA and the basin boundary

12. 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 C t,l, the measure of the riddled basin decreases.

13. 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. 14 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)

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

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

17. 17 Global Effect of Blow-out Bifurcations C Strong synchronizationBubblingRiddling Weight of {PRs} > Weight of {PSs}  SCA  Transversely Unstable Chaotic Saddle  Complete Desynchronization Successive Transverse Bifurcations: Periodic Saddles (PSs)  Periodic Repellers (PRs) (transversely stable) (transversely unstable)  For C < C b,l, absence of an absorbing area  Subcritical blow-out bifurcation Abrupt collapse of the synchronized chaotic state  For C > C b,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  ~  ~ Blow-out Bif. First Transverse Bif. Blow-out Bif.

18. 18 Symmetry-Conserving and -Breaking Blow-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-Conserving Blow-out Bifurcation Symmetry-Breaking Blow-out Bifurcation

19. 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} Hyperchaotic attractor for  =0 Chaotic attractor for  =1 Weight of {PRs} > Weight of {PSs} Weight of {PRs} < Weight of {PSs} Numbers of the period-11 saddles (N s ) and repellers (N r ): N r > N s for  < 0.8 N r 0.9

20. 20 Phase Diagram for The Chaos Synchronization Symmetric coupling (  =0) Dissipatively coupled case with 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 Unidirectional coupling (  =1)

21. 21 First Transverse Bifurcation Their Macroscopic Effects depend on The Existence of The Absorbing Area. 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)  Attractor Bubbling  Basin Riddling  Subcritical case  Abrupt Collapse of A Synchronous Chaotic State  Supercritical case  Appearance of An Asynchronous Chaotic Attractor, Exhibiting The On-Off Intermittency. Chaotic Saddle Weakly-stable SCA Strongly-stable SCA Summary 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. 22 First Transverse Bifurcation Their Macroscopic Effects depend on The Existence of The Absorbing Area. 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)  Attractor Bubbling  Basin Riddling  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. Chaotic Saddle Weakly-stable SCA Strongly-stable SCA Summary 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. 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. 24 Phase Diagram for The Chaos Synchronization Unidirectional coupling (  =1)Symmetric coupling (  =0) Linearly coupled case with 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 Open Circles: Bdry. Crisis of An Absorbing Area, Open Squares: Bdry. Crisis of An Asyn. Chaotic Attractor

25. 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. 26 Phase Diagram for Destruction of Hyperchaotic Attractors

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

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

29. 29 Effect of Parameter Mismatch and Noise for The Bubbling Case 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=C t,r. |y-x| max C t,r  : Mismatching parameter  : Noise strength

30. 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 Abrupt increase of the maximum bursting amplitude is in contrast to the case of symmetric coupling. Symmetric coupling (  =0) |y-x| max Unidirectional coupling (  =1) |y-x| max  ~

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

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