Edward Ott

1. Synchronism in Large Networks of Coupled Heterogeneous Dynamical Systems: Lecture II Edward Ott Generalizations of the Kuramoto Model: External driving and interactions Complex (e.g., chaotic) node dynamics with global coupling Complex node dynamics & network coupling

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

3. Typical Behavior System specified by wi’s and k. Consider N >> 1. g(w)dw = fraction of oscillation freqs. between w and w+dw.

4. Ng∞ = fraction of oscillators whose phases and frequencies lie in the range qto q +dq and w to w +dw

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

6. Examples of generalizations: Interaction with external world Crowd synchrony on the Millennium Bridge A model of circadian rhythm Ref.: Antonsen, Fahih, Girvan, Ott, Platig, arXiv:0711.4135, Chaos (to be published in 9/08) Bridge People Refs.: Eckhardt, Ott, Strogatz, Abrams, McRobie, Phys. Rev. E 75 021110 (2007); Strogatz, et al., Nature (2006).

7. Crowd synchronization on the London Millennium bridge Bridge opened in June 2000

8. The phenomenon: London, Millennium bridge: Opening day June 10, 2000

9. Tacoma narrows bridge Tacoma, Pudget Sound Nov. 7, 1940 See KY Billahm, RH Scanlan, Am J Phys 59, 188 (1991)

10. 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

11. Studies by Arup:

12. Forces during walking: • Downward: mg, about 800 N • forward/backward: about mg • sideways, about 25 N

13. 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)

14. The model Modal expansion for bridge plus phase oscillator for pedestrians: Bridge motion: forcing: phase oscillator:

15. Dynamical simulation

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

17. A Potentially Significant Result Even when the coupled units are chaotic systems that are individually not in any way oscillatory (e.g., 2x 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).

18. 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).

19. Globally Coupled Lorenz Systems

25. Formulation

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

28. Convergence

29. Decay of Mixing Chaotic Attractors kth column Mixing  perturbation decays to zero. (Typically exponentially.)

30. Analytic Continuation • Reasonable assumption  Analytic continuation of : Im(s) Re(s)

31. 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. Conclusion • Framework for the study of networks of heterogeneous dynamical systems coupled on a network. (N >> 1) • Applies to periodic, chaotic and ‘mixed’ ensembles. Our papers can be obtained from : http://www.math.umd.edu/~juanga/umdsyncnets.htm

34. Networks With General Node Dynamics 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). l: Max. eigenvalue of A; depends on network topology (not on node dynamics). Restrepo, Hunt, Ott, PRL ‘06; Physica D ‘06

35. Synchronism in Networks ofCoupled HeterogeneousChaotic (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

36. 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

37. 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