1. 1 Emergence of Coherence and Scaling in an Ensemble of 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. 22 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. 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

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

7. 7 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 Coherence with a Macroscopic Mean Field Successive Appearance of Similar Coherent States 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  Noisy Logistic Map  Parametrically Forced Pendulum [ M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978), J. Crutchfield, M. Nauenberg, and J. Rudnick, Phys. Rev. Lett. 46, 459 (1981). B. Shraiman, C.E. Wayne, and P.C. Martine, Phys. Rev. Lett. 46, 462 (1981).] Universal Scaling Factors:  =4.669 201 …  =-2.502 987 …  =6.619 03  w : Gaussian White Noise with =0 and =  (t 1 – t 2 ). [S.-Y. Kim and K. Lee, Phys. Rev. E 53, 1579 (1996).] -A*-A* h(t)=Acos(2  t) (  =2,  =1, A * = 6.57615 …)

11. 11 Globally Coupled Noisy Chaotic Systems  An Ensemble of Globally Coupled Noisy Logistic Maps A Population of 1D Chaotic Maps Interacting via the Mean Field: Dissipative Coupling Tending to Equalize the States of Elements Investigation: Coherence in Asynchronous Chaotic Attractors and Scaling Associated with the Mean Field Uniform Random Noise with a Zero Mean and Unit Variance Parameter to Control the Noise Strength  Main Interest

12. 12 Scaling near the Zero-Coupling Critical Point (N=10 )  Universal Scaling near the Zero-Coupling Critical Point (a, ,  ) = (a *, 0, 0): (a* = 1.401 155 189 …) Dynamical Behavior at a Set of Parameters (  a, ,  )  Dynamical Behavior [with Doubled Time Scale (t=2  )] at a Set of Renormalized Parameters (  a/ ,  /  c,  /  ) [  =4.669201,  =6.61903] Asynchronous Chaotic Attractors (containing the diagonal) “Scaling Factor for the Coupling Parameter for the Dissipative Coupling: [ 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).] 4

13. 13  Ensemble-Averaged Mean Field h(t) In the Thermodynamic Limit of N→∞, Coherent and Incoherent States Incoherent State (Each Element: Independent Motion) Coherent State (Collective Motion)  Multiple Transitions to Coherence for Incoherent State (Gray, ε=0.02) Coherent State (Black, ε=0.03)

14. 14 Mean- Field Dynamic of Asynchronous Chaotic Attractors for Δa=0.32 ε=0.02 ε=0.03 ε=0.033 ε=0.035ε=0.036 ε=0.035 ε=0.033

15. 15 Order Parameter for the Coherent Transition  Order Parameter Variance (Mean Square Deviation) of the Mean Field h(t) In the Thermodynamic Limit of N→∞, > 0 for the Coherent State → 0 for the Incoherent State ∆ a =0.32, σ=0.001 ε ε*~0.0291 ∆ ε Incoherent StateCoherent State

16. 16 Scaling for the Mean Field Mean Field : Exhibiting the ‘  2 ’-scaling (  =-2.502 987 …) Return Maps of the Mean Field for   a=0.32/  i,  =0.02/2 i,  =0.001/  i, t =2 i  [i (level of renormalization)=0, 1, 2]  Orbital Scalings in the Logistic Map  ‘  ’-scaling near the Critical Point x=0  ‘  2 ’-scaling near the First Iterate of the Critical Point x=1 [= f(0)] x=1 (most concentrated region) x=0 (most rarified region) h(2 τ+1)) τ

17. 17 Appearance of Similar Coherent States of Higher Orders (N=10 )  1st-Order Renormalized State (  a=0.32/ ,  =0.001/  )  2nd-Order Renormalized State (  a=0.32/  2,  =0.001/  2 ) 3rd-Order Renormalized State (  a=0.32/  3,  =0.001/  3 )  =0.039/2 3  =0.039/2 2  =0.039/2 Bifurcation Diagram Variance Diagram Return Map 4

18. 18 Self-Similar State Diagrams (N=10 ) Incoherent States (White) Coherent States (Gray) 1st Renormalized State Diagram 2nd Renormalized State Diagram  =0.001/   =0.001/  2  =0.001 4

19. 19 Effect of Noise on the Transition to Coherence  Scaling for the State Diagram in the  -  Plane  Multiple Transitions to Coherence for  <  * (=0.003)  Single Period-Doubling Transition to Coherence for  >  * 1st Order Renormalized State Diagram  a=0.32/  2nd Order Renormalized State Diagram 0th Order State Diagram  a=0.32/  2

20. 20 A Global Population of Inertially Coupled Maps Renormalization Result for the Scaling of the Coupling Parameter:  Nonlinear Coupling [Tendency of equalizing the states of the elements  Dissipative coupling]  Linear Coupling  One Relevant Scaling Factor: =2  Two Relevant Scaling Factors: 1 =  (=-2.502 987 …): inertial coupling (each element: maintaining the memory of its previous states)  Combination of the Linear and Quadratic Coupling Pure Inertial Coupling with Only One Relevant Scaling Factor: =  [Ref: S.P. Kuznetsov, Chaos, Solitons, and Fractals 2, 281-301 (1992).] [Renormalization Results: S.P.Kuznetsov, Radiophysics and Quantum Electronics 28, 681 (1985). S.-Y. Kim and H. Kook, Phys. Rev. E 48, 785 (1993). S.-Y. Kim and H. Kook, Phys. Lett. A 178, 258 (1993).] 2 = 2

21. 21 Scaling near the Zero-Coupling Critical Point (N=10 )  Universal Scaling near the Zero-Coupling Critical Point (a, ,  ) = (a *, 0, 0): (a* = 1.401 155 189 …) Dynamical Behavior at a Set of Parameters (  a, ,  )  Dynamical Behavior [with Doubled Time Scale (t=2  )] at a Set of Renormalized Parameters (  a/ ,  /  c,  /  ) [  =4.669201,  =6.61903] Asynchronous Chaotic Attractors (containing the diagonal) “Scaling Factor for the Coupling Parameter for the Inertial Coupling: [ 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).] 4

22. 22 Onset of Coherence  Bifurcation Diagram and Return Map of the Mean Field k(t) (N=10 )  Order Parameter Δ a=0.32 σ=0.003 CoherentIncoherentCoherent 4 Coherent State for ε=0.04 Incoherent State for ε=0.02

23. 23 Scaling for the Mean Field Mean Field : Exhibiting the ‘  ’-scaling (  =-2.502 987 …) Return Maps of the Mean Field for   a=0.32/  i,  =0.02/2 i,  =0.003/  i, t =2 i  [i (level of renormalization)=0, 1, 2]  Orbital Scalings in the Logistic Map  ‘  ’-scaling near the Critical Point x=0 x=1 (most concentrated region) x=0 (most rarified region)

24. 24 Scaling Associated with the Mean Field k(t) (N=10 )  1st-Order Renormalized State (  a=0.32,  =0.003)  2nd-Order Renormalized State (  a=0.32/ ,  =0.003/  ) 3rd-Order Renormalized State (  a=0.32/  2,  =0.003/  2 )  =0.04/  2  =0.04/   =0.04 Bifurcation DiagramVariance DiagramReturn Map 4

25. 25 Coherence and Scaling in a Heterogeneous Ensemble of Globally Coupled Maps  Heterogeneous Ensemble (consisting of non-identical elements)  Spread in the map parameter  a i for each element: Randomly chosen with uniform distribution in the interval of (  a- ,  a+  ) (  : spread parameter)  Transition from an Incoherent to a Coherent State Occurrence of a Coherence at  =  * (=0.0289) for  a=0.32 and  =0.006 Coherent State for  =0.03 Incoherent State for ε=0.02

26. 26 Return Map Successive Appearance of Similar Coherent States of Higher Orders  1st-Order Renormalized State (  a=0.32/ ,  =0.006/  )  2nd-Order Renormalized State (  a=0.32/  2,  =0.006/  2 ) 3rd-Order Renormalized State (  a=0.32/  3,  =0.006/  3 ) Bifurcation Diagram Variance Diagram  =0.038/2  =0.038/2 2  =0.038/2 3

27. 27 An Ensemble of Globally Coupled Noisy Pendulums  Globally Coupled Noisy Pendulums  Phase Coherent Attractor in the Single Pendulum  A=0.06  =0.0015  =0.2  =2,  =1  A(  A-A * )=0.06 (A*=6.57615)  =0.0015  Phase Coherence (Flow) and Amplitude Incoherence Mean Field: Rotation on a Noisy Limit Cycle (Flow), Noisy Stationary State (Map)

28. 28 Onset of Coherence  Bifurcation Diagram and Return Map of the Mean Field H(n) (N=10 )  Order Parameter Incoherent Coherent 4 Coherent State for ε=0.04 Incoherent State for ε=0.02

29. 29  1st-Order Renormalized State (  A=0.06/ ,  =0.0015/  )  2nd-Order Renormalized State (  A=0.06/  2,  =0.0015/  2 ) 3rd-Order Renormalized State (  A=0.06/  3,  =0.0015/  3 ) Scaling Associated with the Mean Field H(n) (N=10 ) 4  =0.3/2 2 3 Return MapMSD DiagramBifurcation Diagram

30. 30 Summary  Investigation of Onset of Coherence in an Ensemble of Globally Coupled Noisy Logistic Maps  Universality for the Results Confirmed in a Population of Globally Coupled Noisy Pendulums (a, ,  )  (a *, 0, 0): zero-coupling critical point Successive Appearance of Similar Coherent States of Higher Orders  Our Results: Valid in an Ensemble of Globally Coupled Period-Doubling Systems of Different Nature