dorsal/arxiv
View SchemaAmplitude Expansions for Instabilities in Populations of Globally-Coupled Oscillators
| Authors | John David Crawford |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9310005 |
| URL | https://arxiv.org/abs/patt-sol/9310005 |
| DOI | 10.1007/BF02188217 |
| Journal | J. Stat. Phys. 74, 1047 (1994) |
Abstract
We analyze the nonlinear dynamics near the incoherent state in a mean-field model of coupled oscillators. The population is described by a Fokker-Planck equation for the distribution of phases, and we apply center-manifold reduction to obtain the amplitude equations for steady-state and Hopf bifurcation from the equilibrium state with a uniform phase distribution. When the population is described by a native frequency distribution that is reflection-symmetric 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 Vlasov-Poisson equation. For a bimodal reflection-symmetric distribution, both types of bifurcation are possible and they coincide at a codimension-two Takens Bogdanov point. The steady-state bifurcation may be supercritical or subcritical and produces a time-independent synchronized state. The Hopf bifurcation produces both supercritical stable standing waves and supercritical unstable travelling waves. Previous work on the Hopf bifurcation in a bimodal population by Bonilla, Neu, and Spigler and Okuda and Kuramoto predicted stable travelling waves and stable standing waves, respectively. A comparison to these previous calculations shows that the prediction of stable travelling waves results from a failure to include all unstable modes.
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"abstract": "We analyze the nonlinear dynamics near the incoherent state in a mean-field\nmodel of coupled oscillators. The population is described by a Fokker-Planck\nequation for the distribution of phases, and we apply center-manifold reduction\nto obtain the amplitude equations for steady-state and Hopf bifurcation from\nthe equilibrium state with a uniform phase distribution. When the population is\ndescribed by a native frequency distribution that is reflection-symmetric about\nzero, the problem has circular symmetry. In the limit of zero extrinsic noise,\nalthough the critical eigenvalues are embedded in the continuous spectrum, the\nnonlinear coefficients in the amplitude equation remain finite in contrast to\nthe singular behavior found in similar instabilities described by the\nVlasov-Poisson equation. For a bimodal reflection-symmetric distribution, both\ntypes of bifurcation are possible and they coincide at a codimension-two Takens\nBogdanov point. The steady-state bifurcation may be supercritical or\nsubcritical and produces a time-independent synchronized state. The Hopf\nbifurcation produces both supercritical stable standing waves and supercritical\nunstable travelling waves. Previous work on the Hopf bifurcation in a bimodal\npopulation by Bonilla, Neu, and Spigler and Okuda and Kuramoto predicted stable\ntravelling waves and stable standing waves, respectively. A comparison to these\nprevious calculations shows that the prediction of stable travelling waves\nresults from a failure to include all unstable modes.",
"arxiv_id": "patt-sol/9310005",
"authors": [
"John David Crawford"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1007/BF02188217",
"journal_ref": "J. Stat. Phys. 74, 1047 (1994)",
"title": "Amplitude Expansions for Instabilities in Populations of Globally-Coupled Oscillators",
"url": "https://arxiv.org/abs/patt-sol/9310005"
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