1. 1 Edward Ott University of Maryland Emergence of Collective Behavior In Large Networks of Coupled Heterogeneous Dynamical Systems (Second lecture on network sync)

2. 2 Review of the Onset of Synchrony in the Kuramoto Model (1975) N coupled periodic oscillators whose states are described by phase angle  i, i =1, 2, …, N. All-to-all sinusoidal coupling: Order Parameter;

3. 3 Typical Behavior System specified by  i ’s and k. Consider N >> 1. g(  )d  = fraction of oscillation freqs. between  and  +d .

4. 4 N  ∞N  ∞ = fraction of oscillators whose phases and frequencies lie in the range  to  d  and  to  d 

5. 5 Linear Stability Incoherent state: This is a steady state solution. Is it stable? Linear perturbation: Laplace transform  ODE in  for f  D ( s,k ) = 0 for given g (  ), Re( s ) > 0 implies instability Results: Critical coupling k c. Growth rates. Freqs.

6. 6 Generalizations of the Kuramoto Model General coupling function: Daido, PRL(‘94); sin(θ j -θ i ) f(θ i - θ j ). Time delay: θ j (t) θ j (t- τ ). Noise: Increases k c. ‘Networks of networks’: Communities of phase oscillators are uniformly coupled within communities but have different coupling strengths between communities. Refs.:Barretto, Hunt, Ott, So, Phys.Rev.E 77 03107(2008);Chimera states model of Abrams, Mirollo, Strogatz, Wiley, arXiv0806.0594.

7. 7 A model of circadian rhythm: Crowd synchrony on the Millennium Bridge: Refs.: Eckhardt, Ott, Strogatz, Abrams, McRobie, Phys. Rev. E 75 021110 (2007); Strogatz, et al., Nature (2006). Ref.: Sakaguchi,Prog.Theor.Phys(’88);Antonsen, Fahih, Girvan, Ott, Platig, arXiv:0711.4135;Chaos (to be published in 9/08) Bridge People Generalizations (Continued)

8. 8 Crowd Synchronization on the London Millennium Bridge Bridge opened in June 2000

9. 9 The Phenomenon: London, Millennium bridge: Opening day June 10, 2000

10. 10 Tacoma Narrows Bridge Tacoma, Pudget Sound Nov. 7, 1940 See KY Billahm, RH Scanlan, Am J Phys 59, 188 (1991)

11. 11 Differences Between MB and TB: No resonance near vortex shedding frequency and no vibrations of empty bridge No swaying with few people nor with people standing still but onset above a critical number of people in motion

12. 12 Studies by Arup:

13. 13 Forces During Walking: Downward: mg, about 800 N forward/backward: about mg sideways, about 25 N

14. 14 The Frequency of Walking: People walk at a rate of about 2 steps per second (one step with each foot) Matsumoto et al, Trans JSCE 5, 50 (1972)

15. 15 The Model Bridge motion: forcing: phase oscillator: Modal expansion for bridge plus phase oscillator for pedestrians: (Walkers feel the bridge acceleration through its acceleration.)

16. 16 Dynamical Simulation

17. 17 Coupling complex [e.g.,chaotic] systems Kuramoto model (Kuramoto, 1975) All-to-all Network. Coupled phase oscillators (simple dynamics). Ott et al.,02; Pikovsky et al.96 Baek et al.,04; Topaj et al.01 All-to-all Network. More general dynamics. Ichinomiya, Phys. Rev. E ‘04 Restrepo et al., Phys. Rev E ‘04; Chaos‘06 More general network. Coupled phase oscillators. More general Network. More general dynamics. Restrepo et al. Physica D ‘06

18. 18 A Potentially Significant Result Even when the coupled units are chaotic systems that are individually not in any way oscillatory (e.g., 2 x mod 1 maps or logistic maps), the global average behavior can have a transition from incoherence to oscillatory behavior (i.e., a supercritical Hopf bifurcation).

19. 19 The activity/inactivity cycle of an individual ant is ‘chaotic’, but it is periodic for may ants. Cole, Proc.Roy. Soc. B, Vol. 224, p. 253 (1991).

20. 20 Globally Coupled Lorenz Systems

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26. 26 Formulation

27. 27 Stability of the Incoherent State Goal: Obtain stability of coupled system from dynamics of the uncoupled component

28. 28

29. 29 Convergence

30. 30 Decay of Mixing Chaotic Attractors k th column Mixing  perturbation decays to zero. (Typically exponentially.)

31. 31 Analytic Continuation  Reasonable assumption  Analytic continuation of : Im(s) Re(s)

32. 32 Networks All-to-all : Network : = max. eigenvalue of network adj. matrix An important point: Separation of the problem into two parts:  A part dependent only on node dynamics (finding ), but not on the network topology.  A part dependent only on the network (finding ) and not on the properties of the dynamical systems on each node. 

33. 33

34. 34 Conclusion  Framework for the study of networks of many heterogeneous dynamical systems coupled on a network ( N >> 1 ).  Applies to periodic, chaotic and ‘mixed’ ensembles. Our papers can be obtained from : http://www.chaos.umd.edu/umdsyncnets.html

35. 35 Networks With General Node Dynamics Restrepo, Hunt, Ott, PRL ‘06; Physica D ‘06 Uncoupled node dynamics: Could be periodic or chaotic. Kuramoto is a special case: Main result: Separation of the problem into two parts Q: depends on the collection of node dynamical behaviors (not on network topology).  : Max. eigenvalue of A; depends on network topology (not on node dynamics).

36. 36 Synchronism in Networks of Coupled Heterogeneous Chaotic (and Periodic) Systems Edward Ott University of Maryland Coworkers: Paul So Ernie Barreto Tom Antonsen Seung-Jong Baek Juan Restrepo Brian Hunt http://www.math.umd.edu/~juanga/umdsyncnets.htm

37. 37 Previous Work  Limit cycle oscillators with a spread of natural frequencies: Kuramoto Winfree + many others  Globally coupled chaotic systems that show a transition from incoherence to coherence: Pikovsky, Rosenblum, Kurths, Eurph. Lett. ’96 Sakaguchi, Phys. Rev. E ’00 Topaj, Kye, Pikovsky, Phys. Rev. Lett. ’01

38. 38 Our Work  Analytical theory for the stability of the incoherent state for large ( N >>1) networks for the case of arbitrary node dynamics (  K , oscillation freq. at onset and growth rates).  Examples: numerical exps. testing theory on all- to-all heterogeneous Lorenz systems ( r in [ r -, r + ]).  Extension to network coupling. References: Ott, So, Barreto, Antonsen, Physca D ’02. Baek, Ott, Phys. Rev. E ’04 Restrepo, Ott, Hunt (preprint) arXiv ‘06