Amplitude expansions for instabilities in populations of globallycoupled oscillators
 John David Crawford
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We analyze the nonlinear dynamics near the incoherent state in a meanfield model of coupled oscillators. The population is described by a FokkerPlanck equation for the distribution of phases, and we apply centermanifold reduction to obtain the amplitude equations for steadystate and Hopf bifurcation from the equilibrium state with a uniform phase distribution. When the population is described by a native frequency distribution that is reflectionsymmetric about zero, the problem has circular symmetry. In the limit of zero extrinsic noise, although the critical eigenvalues are embedded in the continuous spectrum, the nonlinear coefficients in the amplitude equation remain finite, in contrast to the singular behavior found in similar instabilities described by the VlasovPoisson equation. For a bimodal reflectionsymmetric distribution, both types of bifurcation are possible and they coincide at a codimensiontwo TakensBogdanov point. The steadystate bifurcation may be supercritical or subcritical and produces a timeindependent synchronized state. The Hopf bifurcation produces both supercritical stable standing waves and supercritical unstable traveling waves. Previous work on the Hopf bifurcation in a bimodal population by Bonilla, Neu, and Spigler and by Okuda and Kuramoto predicted stable traveling waves and stable standing waves, respectively. A comparison to these previous calculations shows that the prediction of stable traveling waves results from a failure to include all unstable modes.
 A. Winfree, Biological rhythms and the behavior of populations of coupled oscillators,J. Theor. Biol. 16:15–42 (1967).
 Y. Kuramoto,Chemical Oscillations, Waves, and Turbulence (SpringerVerlag, New York, 1984).
 H. Sakaguchi, Cooperative phenomena in coupled oscillator system under external fields.Prog. Theor. Phys. 79:39–46 (1988).
 Y. Kuramoto, Cooperative dynamics of oscillator community,Prog. Theor. Phys. Suppl. 79:223–240 (1984).
 Y. Kuramoto and I. Nishikawa, Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator commutities,J. Stat. Phys. 49:569–605 (1987).
 H. Daido, Intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators.J. Stat. Phys. 60:753–800 (1990).
 S. Strogatz and R. Mirollo, Stability of incoherence in a population of coupled oscillators,J. Stat. Phys. 63:613–635 (1991).
 S. Strogatz, R. Mirollo, and P. C. Matthews, Coupled nonlinear oscillators below the synchronization threshold: Relaxation by generalized Landau damping.Phys. Rev. Lett. 68:2730–2733 (1992).
 K. Okuda and Y. Kuramoto, Mutual entrainment between populations of coupled oscillators,Prog. Theor. Phys. 86:1159–1176 (1991).
 H. Daido, Order function and macroscopic mutual entrainment in uniformly coupled limitcycle oscillators,Prog. Theor. Phys. 88:1213–1218 (1992).
 L. L. Bonilla, J. C. Neu, and R. Spigler, Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators,J. Stat. Phys. 67:313–330 (1992).
 G. B. Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies,J. Math. Biol. 22:1–9 (1985).
 J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators,J. Stat. Phys. 72:145–166 (1993).
 E. Knobloch, Bifurcations in rotating systems, inTheory of Solar and Planetary Dynamos: Introductory Lectures, M.R.E. Proctor and A.D. Gilbert, eds. (Cambridge University Press, Cambridge, 1992).
 J. D. Crawford and P. D. Hislop, Application of the method of spectral deformation to the VlasovPoisson system,Ann. Phys. (N.Y.)189:265–317 (1989).
 J. D. Crawford, Amplitude equations on unstable manifolds: Singular behavior from neutral modes, inModern Mathematical Methods in Transport Theory, W. Greenberg and J. Polewczak, eds. (Birkhauser, Basel, 1991), pp. 97–108.
 M. Golubitsky, I. Stewart, and D. G. Schaeffer,Singularities and Groups in Bifurcation Theory, Vol. II (SpringerVerlag, New York, 1988).
 J. D. Crawford and E. Knobloch, Symmetry and symmetrybreaking bifurcations in fluid dynamics,Annu. Rev. Fluid Mech. 23:341–387 (1991).
 M. Reed and B. Simon,Methods of Modern Mathematical Physics, Vol. IV (Academic Press, New York, 1978).
 J. D. Crawford, Introduction to bifurcation theory,Rev. Mod. Phys. 63:991–1037 (1991).
 J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (SpringerVerlag, New York, 1986).
 A. Vanderbauwhede and G. Iooss, Centre manifold theory in infinite dimensions, inDynamics Reported, Vol. 1 (SpringerVerlag, New York, 1992), pp. 125–163.
 D. Ruelle, Bifurcations in the presence of a symmetry group,Arch. Rat. Mech. Anal. 51:136 (1973).
 A. K. Bajaj, Bifurcating periodic solutions in rotationally symmetric systems.SIAM J. Appl. Math. 42:1978–1990 (1982).
 S. A. van Gils and J. MalletParet, Hopf bifurcation and symmetry: Travelling and standing waves on the circle,Proc. R. Soc. Edinburgh 104A:279–307 (1986).
 W. Nagata, Symmetric Hopf bifurcation and magnetoconvection,Contemp. Math. 56:237–265 (1986).
 P. Peplowski and H. Haken, Effects of detuning on Hopf bifurcation at double eigenvalues in laser systems.Phys. Lett. A 120:138–140 (1987).
 G. Dangelmayr and E. Knobloch, The TakensBogdanov bifurcation withO(2) symmetry,Phil. Trans. R. Soc. Lond. A 322:243–279 (1987).
 Title
 Amplitude expansions for instabilities in populations of globallycoupled oscillators
 Journal

Journal of Statistical Physics
Volume 74, Issue 56 , pp 10471084
 Cover Date
 19940301
 DOI
 10.1007/BF02188217
 Print ISSN
 00224715
 Online ISSN
 15729613
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 Oscillators
 bifurcation
 symmetry
 synchronization
 Industry Sectors
 Authors

 John David Crawford ^{(1)}
 Author Affiliations

 1. Department of Physics and Astronomy, University of Pittsburgh, 15260, Pittsburgh, Pennsylvania