Dynamical Routes to Clusters and Scaling in Globally Coupled Chaotic Systems

1. 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) Synchronized Flashing of Fireflies Biological Examples Heartbeats, Circadian Rhythms, Brain Rhythms, Flashing of Fireflies Nonbiological Examples Josephson Junction Array, Multimode Laser, Electrochemical Oscillator Coherent State Incoherent State (each element’s motion: independent) (collective motion) Nonstationary Snapshots Stationary Snapshots t = n+1 t = n 1 (i: index for the element, : Ensemble Average)

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

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

4. 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 4

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

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

7. Synchronization in Pulse-Coupled IF Oscillators  Population of Globally Pulse-Coupled IF Oscillators [Lapicque, J. Physiol. Pathol. Gen. 9, 620 (1971)] Kicking Full Synchronization 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)] 7

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. 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 Transition to Chaos at a Critical Point a* (=1.401 155 189 …) via an Infinite Sequence of Period Doublings : Lyapunov Exponent (exponential divergence rate of nearby orbits)   0  Regular Attractor  > 0  Chaotic Attractor 9

10. Universal Scaling Associated with Period Doublings  Logistic Map Universal Scaling Factors: =4.669 201 … =-2.502 987 … [ M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978),]  Parametrically Forced Pendulum [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) 10

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  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] Investigation: Scaling Associated with Emergence of Clusters 11

12. Complete Chaos Synchronization  Fully Synchronized Chaotic Attractor on the Invariant Diagonal a=0.15 1D Reduced Map Governing the Dynamics of the Fully Synchronized Attractor (FSA):  Transverse Lyapunov Exponent of the FSA : Transverse Lyapunov exponent associated with perturbation transverse to the diagonal a=0.15 For strong coupling, < 0  Complete Synchronization For  < *(~ 0.2901),  > 0  FSA: Transversely Unstable  Transition to Clustering State 12

13. Two-Cluster States  Two-Cluster States on an Invariant 2D Plane 2D Reduced Map Governing the Dynamics of the Two-Cluster State: a=0.15 =0.05 p (=N2/N): Asymmetry Parameter (fraction of the total population of elements in the 2nd cluster) 0 (Unidirectional coupling) < p  1/2 (Symmetric coupling)  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 13

14. Scaling Associated with Periodic Orbits for the Two-Cluster Case  Classification of Periodic Orbits in Terms of the Period and Phase Shift (q,s) q different 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 (180o 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)  Scaling near the Zero-Coupling Critical Point (a*, 0) for p=1/2 Renormalization Results: Scaling Factor for the Coupling Parameter =2 [i.e., ’(=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).] 14

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

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. 16

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

18. Clustering in the Linearly Coupled Maps  System [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)]  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).] 18

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

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

21. 0.10 x 0.05 0.00 2.66 2.74 2.82 A DynamicalRoutestoClustersandScalinginGloballyCoupledOscillators (Purpose: to examine the universality for the results obtained in globally coupled maps) Period-Doubling Route to Chaos in the Single Pendulum (A*=2.759 833) • Globally Coupled Parametrically Forced Pendula (Dissipative Coupling)  Governing Eqs. for the Dynamics of the Two-Cluster States  Scaling for the Conjugate-Phase Periodic Orbits 21

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

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

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