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Low Dimensional Behavior in Large Systems of Coupled Oscillators

Low Dimensional Behavior in Large Systems of Coupled Oscillators. Edward Ott. University of Maryland. References. Main Ref.: E. Ott and T.M. Antonsen, “Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators,”

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Low Dimensional Behavior in Large Systems of Coupled Oscillators

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  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 bepublished in 9/08). • Related Ref.:Antonsen, Faghih, Girvan, Ott and Platig, “External Periodic Driving of Large Systems of Globally Coupled Phase Oscillators,” arXiv: 0711.4135 and Chaos 18 (to be published in 9/08).

  3. Examples of synchronized oscillators • 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.

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

  5. Synchrony in the brain

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

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

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

  9. Order parameter measures the coherence

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

  11. Some Generalizations of The Kuramoto Model: External Drive: drive E.g., circadian rhythm. Ref.: Antonsen, Faghih, Girvan, Ott, Platig, arXiv: 0711.4135,and Chaos (to be published in 9/08). Communities of Oscillators: A = # of communities; σ = community (σ = 1,2,.., s); Nσ = # of individuals in community σ. E.g., chimera states, s= 2 [Abrams, Mirollo, Strogatz, Wiley] .

  12. Generalizations (continued) Time delay:Replace qj(t) by qj(t-t)in the abovegeneralizations. 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→∞ Introduce the distribution function f(q,w,t) [the fraction of oscillators with phases in the range (q,q+dq) and frequencies in the range (w,w+dw) ] 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( w,q,0 ) [or f s( w,q,0 ) in the case of oscillator groups], lying on a submanifold M (specified later) of the space of all possible distribution functions f, • f(w,q, t) continues to lie in M, • for appropriate g(w ) the time evolution of r( t ) (or rs( t )) satisfies a finite set of ODE’s which we obtain. Ott and Antonsen, Chaos (to be published 9/08). *

  16. Comments M • M is an invariant submanifold. • ODE’sgive ‘macroscopic’ evolution of the order parameter. • Evolution of f(w,q,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:0711.4135.

  17. Specifying the Submanifold M The Kuramoto Model as an Example: Inputs:k, the coupling strength, and the initial condition, f(w,q,0) (infinite dimensional). M is specified by two constraints on f(w,q,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 Mis 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)| < 1cannot cross into|α(ω,t)| > 1. |α(ω,t)| < 1and 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)|< 1and . Thus |α| → 0asIm(ω) → -∞for all time t.

  22. Lorentzian g(ω) Set ω = ω0 –iΔin

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

  24. Circadian Rhythm Problem Antonsen et al. • Observed behaviors depending on parameters: • 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 M0= driving strength Ω =frequency mismatch between oscillator average and drive k = 5coupling 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 of N≥ 103ODE’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 ≥ 103ODE’s, • Thus all the observed attractors and bifurcations of the original system occur on M.

  28. A qualitatively similar parameter space diagram applies for Gaussian g(ω). Also, our method can treat certain other g(w)’s, e.g., g(w) ~ [(w-w0 )4 + D4 ]-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. Further Discussion

  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

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