1. 11 Dynamical Routes to Clusters and Scaling in Globally Coupled Chaotic Systems Sang-Yoon Kim Department of Physics Kangwon National University  Globally Coupled Systems (Each element is coupled to all the other ones with equal strength)  Biological Examples Heartbeats, Circadian Rhythms, Brain Rhythms, Flashing of Fireflies  Nonbiological Examples Josephson Junction Array, Multimode Laser, Electrochemical Oscillator Incoherent State (i: index for the element,  : Ensemble Average) (each element’s motion: independent)(collective motion) Coherent State Stationary Snapshots Nonstationary Snapshots t = n t = n+1 Synchronized Flashing of Fireflies

2. 222 Emergent Science “The Whole is Greater than the Sum of the Parts.” Complex Nonlinear Systems: Spontaneous Emergence of Dynamical Order Order Parameter ~ 0 Order Parameter < 1 Order Parameter ~ 1

3. 3 Two Mechanisms for Synchronous Rhythms  Leading by a Pacemaker  Collective Behavior of All Participants

4. 44 Synchronization of Pendulum Clocks Synchronization by Weak Coupling Transmitted through the Air or by Vibrations in the Wall to which They are Attached First Observation of Synchronization by Huygens in Feb., 1665

5. 55 Circadian Rhythms  Biological Clock Ensemble of Neurons in the Suprachiasmatic Nuclei (SCN) Located within the Hypothalamus: Synchronization → Circadian Pacemaker [Zeitgebers (“time givers”): light/dark] Time of day (h) Temperature (ºc) Growth Hormone (ng/mL)

6. 66 Integrate and Fire (Relaxation) Oscillator  Mechanical Model for the IF Oscillator  Van der Pol (Relaxation) Oscillator Accumulation (Integration)“Firing” water level water outflow time  Firings of a Neuron  Firings of a Pacemaker Cell in the Heart

7. 77 Synchronization in Pulse-Coupled IF Oscillators  Population of Globally Pulse-Coupled IF Oscillators Full Synchronization Kicking Heart Beat: Stimulated by the Sinoatrial (SN) Node Located on the Right Atrium, Consisting of Pacemaker Cells [R. Mirollo and S. Strogatz, SIAM J. Appl. Math. 50, 1645 (1990)] [Lapicque, J. Physiol. Pathol. Gen. 9, 620 (1971)]

8. 8 Emergence of Dynamical Order and Scaling in A Large Population of Globally Coupled Chaotic Systems Scaling Associated with Clustering (Partial Synchronization) Successive Appearance of Similar Clusters of Higher Order

9. 99 Period-Doubling Route to Chaos  Lorenz Attractor [Lorenz, J. Atmos. Sci. 20, 130 (1963)] Butterfly Effect [Small Cause  Large Effect] Sensitive Dependence on Initial Conditions  Logistic Map [May, Nature 261, 459 (1976)] : Representative Model for Period-Doubling Systems  : Lyapunov Exponent (exponential divergence rate of nearby orbits)   0  Regular Attractor  > 0  Chaotic Attractor Transition to Chaos at a Critical Point a * (=1.401 155 189 …) via an Infinite Sequence of Period Doublings

10. 10 Universal Scaling Associated with Period Doublings  Logistic Map  Parametrically Forced Pendulum [ M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978),] Universal Scaling Factors:  =4.669 201 …  =-2.502 987 … [S.-Y. Kim and K. Lee, Phys. Rev. E 53, 1579 (1996).] h(t)=Acos(2  t) (  =0.7,  =1.0, A * =2.759 832)

11. 11 Globally Coupled Chaotic Maps  An Ensemble of Globally Coupled Logistic Maps A Population of 1D Chaotic Maps Interacting via the Mean Field: Dissipative Coupling Tending to Equalize the States of Elements Investigation: Scaling Associated with Emergence of Clusters  Main Interest Occurrence of Clustering (Appearance of Clusters with Different Synchronized Dynamics) [Experimental Observations: Electrochemical Oscillators, Salt-Water Oscillators, Belousov-Zhabotinsky Reaction, Catalytic CO Oxidation]

12. 12  Fully Synchronized Chaotic Attractor on the Invariant Diagonal Complete Chaos Synchronization  Transverse Lyapunov Exponent of the FSA 1D Reduced Map Governing the Dynamics of the Fully Synchronized Attractor (FSA): : Transverse Lyapunov exponent associated with perturbation transverse to the diagonal For strong coupling,   < 0  Complete Synchronization For  0  FSA: Transversely Unstable  Transition to Clustering State  a=0.15

13. 13  Two-Cluster States on an Invariant 2D Plane Two-Cluster States  Transverse Lyapunov Exponents  ,1 (  ,2 ): Transverse Lyapunov exponent associated with perturbation breaking the synchrony of the 1st (2nd) cluster  ,1 <0 and  ,2 <0  Two-cluster state: Transversely Stable  Attractor in the original N-D state space 2D Reduced Map Governing the Dynamics of the Two-Cluster State: p (=N 2 /N): Asymmetry Parameter (fraction of the total population of elements in the 2nd cluster ) 0 (Unidirectional coupling) < p  1/2 (Symmetric coupling)  a=0.15  =0.05

14. 14  Classification of Periodic Orbits in Terms of the Period and Phase Shift (q,s) Scaling Associated with Periodic Orbits for the Two-Cluster Case  Scaling near the Zero-Coupling Critical Point (a *, 0) for p=1/2  q d ifferent orbits with period q distinguished by the phase shift s (=1,…,q-1) in the two uncoupled (  =0) logistic maps  (Synchronous) In-phase orbit on the diagonal (s = 0)  (Asynchronous) Anti-phase (180 o out-of-phase) orbit with time shift of half a period (s = q/2)  (Asynchronous) Non-antiphase orbits (Other s) Two orbits with phase shifts s and q- s: Conjugate-phase orbits (under the exchange X Y for p=1/2) Stability Diagrams of the Conjugate-Phase Periodic Orbits [1. S.P.Kuznetsov, Radiophysics and Quantum Electronics 28, 681 (1985). 2. S.-Y. Kim and H. Kook, Phys. Rev. E 48, 785 (1993). 3. S.-Y. Kim and H. Kook, Phys. Lett. A 178, 258 (1993).] Renormalization Results: Scaling Factor for the Coupling Parameter  =2 [i.e.,  ’ (=2  )]

15. 15 Dynamical Routes to Two-Cluster States (p=1/2)  Dynamical Route to Two-Cluster State for  a=0.15 (Two Stages) FSA (Strong coupling) Blowout Bifurcation (  * =0.2901) (“Complex” Gray line) Transversely Unstable (1) Jump to Anti-phase Period-2 Two-Cluster State  Complete Synchronization Stabilization of anti-phase period-2 attractor via subcritical pitchfork bifurcation (2) Transition to Conjugate-Phase Period-4 Two-Cluster States For  < 0.0862, Two-Cluster Chaotic State: Transversely Unstable (Gray dots)  High-Dimensional State

16. 16 Scaling for the Dynamical Routes to Clusters (p=1/2)  Successive Appearance of Similar Cluster States of Higher Orders (1) 1st-Order Renormalized State (2) 2nd-Order Renormalized State  As the zero-coupling critical point (a *, 0) is approached, similar cluster states of higher orders appear successively.

17. 17 Effect of Asymmetric Distribution of Elements p (asymmetry parameter): smaller  Conjugate-Phase Two-Cluster States: Dominant  Appearance of Similar Cluster States

18. 18  System Clustering in the Linearly Coupled Maps  Governing Eqs. for the Two-Cluster State  Scaling for the Linear Coupling Case (P=1/2) [  for the inertial coupling case;  2  for the dissipative coupling case] [ Renormalization Results: 1. S.P.Kuznetsov, Radiophysics and Quantum Electronics 28, 681 (1985). 2. S.-Y. Kim and H. Kook, Phys. Rev. E 48, 785 (1993). 3. S.-Y. Kim and H. Kook, Phys. Lett. A 178, 258 (1993).] [Linear Mean Field  ‘Inertial Coupling’ (each element: maintaining the memory of its previous states)] Nonlinear Mean Field  Dissipative Coupling (Tendency of equalizing the states of the elements)]

19. 19 Successive Appearance of Similar Cluster States of Higher Orders for the Linear Coupling Case (P=1/2) (1) 0th-Order Cluster State (2) 1st-Order Renormalized State (3) 2nd-Order Renormalized State (No Complete Chaos Synchronization near the Zero Coupling Critical Point)

20. 20 Asymmetric Effect on the Dynamical Routes to Clusters p (asymmetry parameter): smaller  Conjugate-Phase Two-Cluster States: Dominant  Appearance of Similar Cluster States (Similar to the Dissipative Coupling Case)

21. 21 Dynamical Routes to Clusters and Scaling in Globally Coupled Oscillators (Purpose: to examine the universality for the results obtained in globally coupled maps)  Globally Coupled Parametrically Forced Pendula (Dissipative Coupling)  Governing Eqs. for the Dynamics of the Two-Cluster States  Scaling for the Conjugate-Phase Periodic Orbits 2.662.742.82 0.00 0.05 0.10 A x Period-Doubling Route to Chaos in the Single Pendulum (A * =2.759 833)

22. 22 Similar Cluster States for the Dissipative Coupling Case (1) 0th-Order Cluster State (2) 1st-Order Renormalized State (3) 2nd-Order Renormalized State

23. 23 Similar Clusters for the Inertial Coupling Case (2) 1st-Order Renormalized State  System  Appearance of Similar Cluster States (Scaling for the Coupling Parameter:    ) (1) 0th-Order Cluster State (3) 2nd-Order Renormalized State

24. 24 Similar Clusters in Globally Coupled Rössler Oscillators (1) 0th-Order Cluster State (2) 1st-Order Renormalized State  Globally Coupled Rössler Oscillators (Dissipative Coupling)  Appearance of Similar Cluster States

25. 25 Summary  Investigation of Dynamical Routes to Clusters in Globally Coupled Logistic Maps  Universality for the Results Confirmed in Globally Coupled Pendulums (a, c)  (a *, 0): zero-coupling critical point Successive Appearance of Similar Cluster States of Higher Orders  Our Results: Valid in Globally Coupled Period-Doubling Systems of Different Nature