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

2 References Main Refs.: E. Ott and T.M. Antonsen, “Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators,” arXiv: and Chaos 18, (‘08). “Long Time Behavior of Phase Oscillator Systems”, arXiv: (‘09) and Chaos 19, (‘09). Our other related work that is referred to in this talk can be found at:

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

4 Cellular clocks in the brain (day-night cycle). Yamaguchi et al., Science 302, 1408 (‘03). 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. 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 max 

11 External Drive: Generalizations of the Kuramoto Model drive E.g., circadian rhythm. Ref.: Sakaguchi, ProgTheorPhys(‘88); Antonsen, Faghih, Girvan, Ott, Platig,Chaos 18 (‘08); Childs, Strogatz, Chaos 18 (‘08). Communities of Oscillators: A = # of communities; σ = community (σ = 1,2,.., s); N σ = # of individuals in community σ. E.g., Barreto, et al., PhysRevE 77 (’08);Martens, et al., PhysRevE (‘09)(two humped g(ω), s=2); Abrams,et al.,PhysRevLett 101(‘08); Laing, Chaos19(’09); and Pikovsky&Rosenblum, PhysRevLett 101 (‘08); Sethia, Strogatz, Sen, SIAM meeting (‘09).

12 Generalizations (continued) Millennium Bridge Problem: ( Bridge mode) ( Walker force on bridge) ( Walker phase) Ref.: Eckhardt,Ott,Strogatz,Abrams,McRobie, PhysRevE 75, (‘07); Abdulrehem and Ott, Chaos 19, (‘09).

13 Generalizations (continued) Time varying coupling: The strengths of the interaction between oscillators depends on t. Ref.: So, Cotton, Barreto, Chaos 18 (‘08). Time delays in the connections between oscillators: Each link can have a different delay with the collection of delays characterized by a distribution function. Ref.: W.S.Lee, E.Ott, T.M.Antonsen, arXiv: (PhysRevLett, to be published). Josephson junction circuits: Ref.: S.A.Marvel, S.H.Strogatz, Chaos 18, (2009). But there was a puzzle.

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

15 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

16 The Main Message of This Lecture* 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, arXiv: and Chaos 18, (‘08); and arXiv: (’09) and Chaos 19, (‘09). *

17 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: we have recently shown that, under weak conditions, M contains all the attractors and bifurcations of the order parameter dynamics when there is nonzero spread in the frequency distribution. M

18 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):

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

20 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. The problem for α(ω,t) is still infinite dimensional.

21 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 ).

22 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.

23 Lorentzian g( ω ) Set ω = ω 0 –iΔ in

24 Solution for |R(t)|=r(t) Thus this steady solution is nonlinearly stable and globally attracting.

25 Crowd Synchronization on the London Millennium Bridge Bridge opened in June 2000

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

27 Studies by Arup:

28 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)

29 MODEL REDUCED MODEL Model expansion for bridge + phase oscillators for walkers Ref.:Eckhardt, Ott, Strogatz, Abrams and McRobie, Phys.Rev.E75, (‘07) Ref.: M.M.Abdulrehem and E.Ott, Chaos 19, (‘09), arXiv: F(ω) ω 

30 NUMERICAL SOLUTIONS OF REDUCED EQS.

31 Further Discussion Our method can treat certain other g(  )’s, e.g., g(  ) ~ [(   ) 4 +  4 ] -1, or g(ω)=[polynomial]/[polynomial]. Then there are s coupled ODE’s for s order parameters where s is the number of poles of g(ω) in Im(ω)<0. For generalizations in which s interacting groups are treated our method yields a set of s coupled complex ODE’s for s complex order parameters. E.g., s=2 for the chimera problem. Numerical and analytical work [e.g., Martens, et al. (‘09); Lee, et al. (‘09); Laing (‘09)] shows similar results from Lorentzian and Gaussian distributions of oscillator frequencies, implying that qualitative behavior does not depend on details of g(ω).

32 ATTRACTION TO M Ott & Antonsen have recently rigorously shown that, for the Kuramoto model and its generalizations discussed above, all attractors of the order parameter dynamics and their bifurcations occur on M, provided that  > 0, and certain weak additional conditions are satisfied. I.e., M is an inertial manifold wrt a proper choice of the distance metric in the space of distribution functions f. Ref.: arXiv: ; and Chaos 19, (‘09). If  =0, long time behavior not on M can occur [e.g., Pikovsky & Rosenblum, PhysRevLett (‘08); Marvel & Strogatz, Chaos 19 (‘09); Sethia, Strogatz & Sen, SIAM mtg.(‘09)]. The long time behavior of systems of heterogeneous oscillators is simpler when the oscillator frequencies are heterogeneous (  >0). (Furthermore, for all the previous generalizations of Kuramoto,  >0 is the more realistic model.)

33 TRANSIENT BEHAVIOR: ECHOES The transient behavior that occurs as the orbit relaxes to M can be nontrivial. An example of this is the ‘echo’ phenomenon studied in Ott, Platig, Antonsen & Girvan, Chaos 18, (‘08). [Similar to Landau echoes in plasmas; e.g., T.M.O’Neil & R.W.Gould, Phys. Fluids (1968).] For the classical Kuramoto model with k below its critical value and external stimuli: k < k c r t Stimulus Echo

34 AN ANALOGY N eqs. for N>>1 oscillator phases. Relaxation to M. ODE description for order parameter. Hamilton’s eqs. for N>>1 interacting fluid particles. Relaxation to a local Maxwellian. Fluid eqs. for moments (density, velocity, temp., …).

35 Conclusion The long time macroscopic behavior of large systems of globally coupled oscillators has been demonstrated to be low dimensional. Systems of ODE’s describing this low dimensional behavior can be explicitly obtained and utilized to discover and analyze all the long time behavior (e.g., the attractors and bifurcations) of these systems. Ref.: Ott & Antonsen, Chaos 18, (2008). Also, for the demonstration of attraction to M see Chaos 19, (2009).