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Synchronization in Coupled Chaotic Systems

Synchronization in Coupled Chaotic Systems. Sang-Yoon Kim Department of Physics Kangwon National University. Synchronization in Coupled Periodic Oscillators. Synchronous Pendulum Clocks. Synchronously Flashing Fireflies. Chaos and Synchronization. [Lorenz, J. Atmos. Sci. 20 , 130 (1963).].

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Synchronization in Coupled Chaotic Systems

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

  2. Chaos and Synchronization [Lorenz, J. Atmos. Sci. 20, 130 (1963).] Butterfly Effect: Sensitive Dependence on Initial Conditions [Small Cause  Large Effect] z •  Lorenz Attractor y x Coupled Chaotic (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).] (Secret Message) Chaotic System Chaotic System  + - Transmitter Receiver  Encoding by Using Chaotic Masking  Decoding by Using Chaos Synchronization

  4. 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).]

  5. Coupled 1D Maps  1D Map (Representative model exhibiting universal scaling behavior) (x: seasonly breeding inset population) Iterates: (trajectory)  Attractor Period-Doubling Transition to Chaos When a exceed a, a chaotic attractor with a positive Lyapunov exponent s appears. An infinite sequence of period doubling bifurcations ends at a finite accumulation point   ( > 0  chaotic attractor,  < 0  regular attractor) a a a a

  6.  Two Coupled 1D Maps Coupling Function  c: coupling parameter Asymmetry Parameter  (0    1)   = 0: symmetric coupling  exchange symmetry   0: asymmetric coupling ( = 1: unidirectional coupling)  Invariant Synchronization Line y = x Synchronous Orbits on the Diagonal (, ) Asynchronous Orbits off the Diagonal () 

  7. Transverse Stability of the Synchronous Chaotic Attractor Synchronous Chaotic Attractor (SCA) on the Invariant Diagonal • SCA: Stable against the “Transverse Perturbation”  Chaos Synchronization • An Infinite Number of Unstable Periodic Orbits (UPOs) Embedded in the SCA and Forming Its Skeleton  Intimately Associated with the Transverse Stability of the SCA

  8. Competition between Periodic Saddles and Repellers  Investigation of Transverse Stability of the SCA in Terms of UPOs (: transverse Lyapunov exponent of the SCA) {UPOs} = {Transversely Stable Periodic Saddles (PSs)} + {Transversely Unstable Periodic Repellers (PRs)}  Chaos Synchronization “Strength” of {PSs} > “Strength” of {PRs}   < 0 (SCA: transversely stable)  Complete Desynchronization “Strength” of {PSs} < “Strength” of {PRs}   > 0 (SCA: transversely unstable chaotic saddle) Chaos Synchronization c Blowout Bifurcation Blowout Bifurcation

  9. LossofChaosSynchronization Unidirectionally Coupled 1D Maps (=1) c: Coupling Parameter State Diagram • Appearance of a Synchronous Chaotic Attractor (SCA) on the Invariant Diagonal when Passing a Critical Line (heavy solid line). a=1.82 • An Infinite Number of UPOs inside the SCA. a=1.401 155 … •Strong Synchronization (Hatched Region, <0) All Synchronous UPOs: Transversely Stable Periodic Saddles  No Bursting (attracted to the diagonal without any bursting) • Transition to Weak Synchronization (Gray and Dark Gray Regions, <0) via a First Transverse Bifurcation of a Periodic Saddle Some UPOs: Transversely Unstable  Local Bursting

  10. Fate of Local Bursting for the Case of Weak Synchronization Weak Synchronization (Riddling) Weak Synchronization (Bubbling) Strong Synchronization c First Transverse Bifurcation First Transverse Bifurcation Blowout Bifurcation Blowout Bifurcation •Weak Synchronization: Some UPOs: Transversely Unstable  Local Bursting  Fate of Local Bursting Dependent on the Global Dynamics Attractor Bubbling (Gray Region) Folded Back of a Locally Repelled Trajectory  Transient Intermittent Bursting (<0) Basin Riddling (Dark Gray Region) Basin of the SCA: Riddled with a Dense Set of “Holes,” Leading to Another Attractor Attracted to Another Distant Attractor

  11. Transcritical Transition to Basin Riddling  Transcritical Bifurcation (Stability Exchange) [S.-Y. Kim and W. Lim, Phys. Rev. E 63, 026217 (2001). S.-Y. Kim and W. Lim, Prog. Theor. Phys. 105, 187 (2001). S.-Y. Kim and W. Lim, Phys. Rev. E 64, 016211 (2001).] Riddling Strong synchronization Bubbling c Supercritical Period-Doubling Bif. Transcritical Bif.  Riddling Transition (Basin: riddled with a dense set of “holes”)  an absorbing area surrounding the SCA (: repeller on the basin boundary : saddle on the diagonal) Contact between the SCA and the basin boundary  Riddled Basin After the transcritical bifurcation, the basin becomes “riddled” with a dense set of “holes” leading to divergent orbits. As c decreases from ct,l, the basin riddling is intensified.

  12. Characterization of the Riddled Basin  Divergence Exponent Divergence Probability P(d) ~ d Measure of the Basin Riddling [c  Blowout Bifurcation Point cb,l (=-2.963)  ()  P(d): Increase] c  Uncertainty Exponent Uncertainty Probability P() ~  Two Initial Condition: Uncertain if their final states are different  Fine Scaled Riddling of the SCA [c  cb,l  ()  P(): Increase] c

  13. Effect of Parameter Mismatching on Weak Synchronization [A. Jalnin and S.-Y. Kim, Phys. Rev. E 65, 026210 (2002). S.-Y. Kim, W. Lim, and Y. Kim, Prog. Theor. Phys. 107, 239 (2002).]  Unidirectionally Coupled Nonidentical 1D Maps  : Mismatching Parameter  Effect of Parameter Mismatching (=0.001) (1) Strong Synchronization (Small Mismatching  Small Effect) Slightly Perturbed SCA ( Mismatching Strength) All UPOs (embedded in the SCA): Transversely Stable  No Parameter Sensitivity (2) Weak Synchronization (Small Mismatching  Large Effect) Local Transverse Repulsion of Periodic Repellers  Parameter Sensitivity  Riddling  Bubbling SCA with a Riddled Basin (Gray)  Chaotic Transient (Black) with a Finite Lifetime SCA  Bubbling Attractor (exhibiting persistent intermittent bursting)

  14. Characterization of the Parameter Sensitivity of a Weakly Stable SCA  Characterization of Parameter Sensitivity • Measured by Calculating a Derivative of the Transverse Variable un (=xn-yn) with respect to the Mismatching Parameter along a Synchronous Trajectory  Boundedness of SN Intermittent Behavior Looking only at the Maximum Values of |SN|: • Representative Value (by Taking the Minimum Value of in an Ensemble of Randomly Chosen Initial Orbit Points) Parameter Sensitivity Function: • Strong Synchronization (SS) N: Bounded  No Parameter Sensitivity •Weak Synchronization (WS) N ~ N: Unbounded  Parameter Sensitivity : Parameter Sensitivity Exponent (PSE) Used to Measure the Degree of Parameter Sensitivity c=-0.7 (WS) c=-1.5 (SS)

  15. Characterization of the Bubbling Attractor and the Chaotic Transient a=1.82, c=-0.7, =0.001 0.8 un 0.0 -0.8 0 3000 1500 n  Parameter Sensitivity Exponents (PSEs) • Monotonic Increase of  () as c is Changed toward the Blowout Bifurcation Point (Due to the Increase in the Strength of Local Transverse Repulsion of the Embedded Periodic Repellers.) • Scaling for the Average Characteristic Time • Average Laminar Length (Interburst Interval) of the Bubbling Attractor:  ~  -  Average Lifetime of Chaotic Transient:  ~  - • Reciprocal Relation between the Scaling Exponent  and the PSE   ~ -1/   () =1/ ()

  16. Effect of Noise on Weak Synchronization [S.-Y. Kim, W. Lim, A. Jalnin, and S.-P. Kuznetsov, Phys. Rev. E 67, 016217 (2003).]  Unidirectionally Coupled Noisy 1D Maps  Characterization of the Noise Sensitivity (=0.0005) Chaotic Transient Bubbling Attractor : Bounded Noise → Boundedness of SN: Determined by RM (same as in the parameter mismatching case) • Characterization of the Bubbling Attractor and the Chaotic Transient  ~  - ;  () =1/ () (: average time spending near the diagonal) [Noise Sensitivity Exponent() = PSE()] Noise Effect = Parameter Mismatching Effect

  17. Dynamical Consequence of Blowout Bifurcations [S.-Y. Kim, W. Lim, E. Ott, and B. Hunt, Phys. Rev. E 68, 066203 (2003).]  Two Coupled 1D maps : parameter tuning the degree of asymmetry of the coupling =0  Symmetric Coupling Case, =1  Unidirectional Coupling Case Synchronization c Blowout Bifurcation Blowout Bifurcation  Appearance of an Asynchronous Attractor via a Blowout Bifurcation of the SCA Asynchronous Hyperchaotic Attractor with 2>0 for =0 Asynchronous Chaotic Attractor with 2<0 for =1

  18. Mechanism for the Appearance of the Asynchronous Hyperchaotic and Chaotic Attractors • Decomposition of the 2nd Lyapunov Exponent 2 d = |v| [|(x-y)/2|]: Transverse Variable d < d*: Laminar Component (Off State) d > d*: Bursting Component (On State) Intermittent Asynchronous Attractor Born via a Blowout Bifurcation d (t) Weighted 2nd Lyapunov Exponent for the Laminar (Bursting) Component d* • Competition between the Laminar and Bursting Components : =0, : =0.852  : =1 Sign of 2 (= b2- |l2|): Determined by the Competition between the Laminar and Bursting Components

  19. Partial Synchronization in Three Coupled Chaotic Systems [W. Lim and S.-Y. Kim, Phys. Rev. E 71, 035221 (2005).] • Three Coupled 1D Maps • Occurrence of the Partial Synchronization  Fully Synchronized Attractor (FSA) for the Case of Strong Coupling Breakdown of the Full Synchronization via a Blowout Bifurcation Partial Synchronization (PS) Complete Desynchronization : Two-Cluster State for p=0

  20. Transverse Stability of Two-Cluster States  Reduced 2D Map Governing the Dynamics of a Two-Cluster State Two-Cluster State:  Intermittent Two-Cluster State Born via the Blowout Bifurcation of the FSA Unidirectional Coupling Case Symmetric Coupling Case  Threshold Value p* (~ 0.146) s.t. • 0p<p*  Two-Cluster State: Transversely Stable (<0)  Occurrence of the PS •p*<p1/3  Two-Cluster State: Transversely Unstable (>0)  Occurrence of the Complete Desynchronization

  21. Mechanism for the Occurrence of the Partial Synchronization • Decomposition of the Transverse Lyapunov Exponent  d = |V| [V(X-Y)/2]: Transverse Variable d < d*: Laminar Component, d > d*: Bursting Component. Intermittent Two-Cluster State Born via a Blowout Bifurcation d (t) Weighted Transverse Lyapunov Exponent for the Laminar (Bursting) Component d* • Competition between the Laminar and Bursting Components : p=0, : p=0.146 : p=1/3 Sign of  (= b- |l|): Determined by the Competition between the Laminar and Bursting Components

  22. Weak Synchronization (Riddling) Weak Synchronization (Bubbling) Strong Synchronization Blowout Bifurcation First Transverse Bifurcation First Transverse Bifurcation Blowout Bifurcation Summary  Investigation of Loss of Chaos Synchronization in Two Coupled Chaotic Systems 1.Transcritical Transition to Basin Riddling in Asymmetrically Coupled Chaotic Systems 2. Characterization of the Parameter Mismatching and Noise Effects on Weak Synchronization in terms of the PSEs and NSEs 3. Investigation of Dynamical Origin for the Occurrence of Hyperchaos and Chaos via Blowout Bifurcation through Competition between the Laminar and Bursting Components  Investigation of Loss of Full Synchronization in Three Coupled Chaotic Systems Partial Synchronization (Clustering) Full Synchronization Complete Desynchronization 1. Investigation of the Dynamical Mechanism for the Occurrence of the Partial Synchronization through Competition between the Laminar and Bursting Components

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