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 Synchrony in 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:
Order Parameter;

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. N g ∞
= fraction of oscillators whose phases and frequencies lie in the range q to 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
A model of circadian rhythm
Ref.: Antonsen, Fahih, Girvan, Ott, Platig, arXiv: , Chaos (to be published in 9/08)
Crowd synchrony on the Millennium Bridge
Bridge
People
Refs.: Eckhardt, Ott, Strogatz, Abrams, McRobie, Phys. Rev E (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 dynamics.
More general
network.
Coupled phase
oscillators.
Ott et al.,02; Pikovsky et al.96
Baek et al.,04; Topaj et al.01
Ichinomiya, Phys. Rev. E ‘04
Restrepo et al., Phys. Rev E ‘04; Chaos‘06
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 :

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. Edward Ott University of Maryland
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

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 ’ Baek, Ott, Phys. Rev. E ’ Restrepo, Ott, Hunt (preprint) arXiv ‘06