1 Low Dimensional Behavior in Large Systems of Coupled Oscillators Edward Ott University of Maryland

2 References Main Ref.: E. Ott and T.M. Antonsen, “Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators,” Chaos 18 (to be published in 9/08). Related Ref.: Antonsen, Faghih, Girvan, Ott and Platig, “External Periodic Driving of Large Systems of Globally Coupled Phase Oscillators,” arXiv: and Chaos 18 (to be published in 9/08).

3 Cellular clocks in the brain. Pacemaker cells in the heart. Pedestrians on a bridge. Electric circuits. Laser arrays. Oscillating chemical reactions. Bubbly fluids. Neutrino oscillations. Examples of synchronized oscillators

4 Cellular clocks in the brain (day-night cycle). Yamaguchi et al., Science, vol.302, p.1408 (2003). Incoherent Coherent

5 Synchrony in the brain

6 Coupled phase oscillators  Change of variables Limit cycle in phase space Many such ‘phase oscillators’: Couple them: Kuramoto: ; i=1,2,…,N »1 Global coupling

7 Framework N oscillators described only by their phase . N is very large. g(  )  The oscillator frequencies are randomly chosen from a distribution g(  ) with a single local maximum. (We assume the mean frequency is zero) nn nn

8 Kuramoto model (1975) Macroscopic coherence of the system is characterized by n = 1, 2, …., N k= (coupling constant) = “order parameter”

9 Order parameter measures the coherence

10 Results for the Kuramoto model There is a transition to synchrony at a critical value of the coupling constant. r Incoherence Synchronization g(0) g(  ) 

11 External Drive: Some Generalizations of The Kuramoto Model: drive E.g., circadian rhythm. Ref.: Antonsen, Faghih, Girvan, Ott, Platig, arXiv: , and Chaos (to be published in 9/08). Communities of Oscillators: A = # of communities; σ = community (σ = 1,2,.., s); N σ = # of individuals in community σ. E.g., chimera states, s = 2 [Abrams, Mirollo, Strogatz, Wiley].

12 Generalizations (continued) Time delay: Replace  j (t) by  j (t-  in the above generalizations. Millenium Bridge Problem: ( Bridge mode) ( Walker force on bridge) ( Walker phase) Ref.: Eckhardt, Ott, Strogatz, Abrams, McRobie, Phys. Rev. E (2007).

13 The ‘Order Parameter’ Description “The order parameter”

14 N→∞N→∞ Introduce the distribution function f( ,t) [the fraction of oscillators with phases in the range (  +d  ) and frequencies in the range (  +d  ) ] Conservation of number of oscillators: 0 and similar formulations for generalizations

15 The Main Result* Considering the Kuramoto model and its generalizations, for i.c.’s f( ,0 ) [or f  ,0 ) in the case of oscillator groups], lying on a submanifold M (specified later) of the space of all possible distribution functions f, f( , t) continues to lie in M, for appropriate g(  ) the time evolution of r( t ) (or r  ( t )) satisfies a finite set of ODE’s which we obtain. Ott and Antonsen, Chaos (to be published 9/08). 

16 Comments M is an invariant submanifold. ODE’s give ‘macroscopic’ evolution of the order parameter. Evolution of f( ,t ) is infinite dimensional even though macroscopic evolution is finite dimensional. Is it useful? Yes, if the dynamics of r(t) found in M is attracting in some sense. Ref.: Antonsen, Faghih, Girvan, Ott, Platig, Chaos (9/08), arXiv: M

17 Specifying the Submanifold M The Kuramoto Model as an Example: Inputs: k, the coupling strength, and the initial condition, f (  (infinite dimensional). M is specified by two constraints on f(  0):

18 Specifying the Manifold M (continued) Fourier series for f: Constraint #1: ? Question: For t >0 does

19 Specifying the Manifold M (continued) Constraint #2: α(ω,0) is analytic for all real ω, and, when continued into the lower-half complex ω-plane ( Im(ω)< 0 ), (a) α(ω,0) has no singularities in Im(ω)< 0, (b) lim α(ω,0) → 0 as Im(ω) → - ∞. It can be shown that, if α(ω,0) satisfies constraints 1 and 2, then so does α(ω,t) for all t < ∞. The invariant submanifold M is the collection of distribution functions satisfying constraints 1 and 2.

20 If, for Im( ω )< 0, | α ( ω,0)|<1, then | α ( ω,t)|<1 : Multiply by α * and take the real part: At |α(ω,t)|=1: |α| starting in |α(ω,0)| 1. |α(ω,t)| < 1 and the solution exists for all t ( Im(ω)< 0 ).

21 If α ( ω,0) → 0 as Im( ω ) → -∞, then so does α ( ω,t) Since |α| < 1, we also have (recall that ) | R(t)|< 1 and. Thus |α| → 0 as Im(ω) → -∞ for all time t.

22 Lorentzian g(ω) Set ω = ω 0 –iΔ in

23 Solution for |R(t)|=r(t)

24 Circadian Rhythm Problem Antonsen et al. Observed behaviors depending on parameters: A.One globally attracting state in which the drive entrains the oscillator system. B. An unentrained state is the attractor (bad sleep pattern). C. Same as in A, but there are also two additional unstable entrained solutions.

25 Parameter Space M 0 = driving strength Ω = frequency mismatch between oscillator average and drive k = 5 coupling strength

26 Schematic Blow-up Around T A↔B: Hopf bifurcation A ↔ C: Saddle-node bifurcation of 2 and 3 C ↔ B: Saddle-node bifurcation on a periodic orbit (1 and 2 created as B → C)

27 Low Dim. ODE Reduction of the Circadian Rhythm Model The above results were obtained from solution of the full problem for f(ω,θ,t) (without restricting the dynamics to M ), e.g., partly by numerical solution ofN ≥ 10 3 ODE’s, The ODE for evolution on the manifold M is The results from solution of this equation for give the same picture (quantitatively!) as obtained from solving the N ≥ 10 3 ODE’s, Thus all the observed attractors and bifurcations of the original system occur on M.

28 Further Discussion A qualitatively similar parameter space diagram applies for Gaussian g(ω). Also, our method can treat certain other g(  )’s, e.g., g(  ) ~ [(   ) 4 +  4 ] -1. Numerical studies of other generalizations of the Kuramoto model (e.g., chimera states [Abrams, Mirollo, Strogatz, Wiley]) also show all the interesting dynamics taking place on M. For generalizations of the Kuramoto problem in which s interacting groups are treated (e.g., s=2 for the chimera problem), our method yields a set of s coupled complex ODE’s for s complex order parameters describing the system’s state. For the Millenium Bridge model we get a 2nd order ODE for the bridge driven by a complex order parameter describing the collective state of the walkers, plus an ODE for the walker order parameter driven by the bridge.

29 Conclusion The macroscopic behavior of large systems of globally coupled oscillators have been demonstrated (at least in some cases) to be low dimensional. Thanks: Tom Antonsen Michelle Girvan Rose Faghih John Platig Brian Hunt