Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Amplitude Expansions for Instabilities in Populations of
  Globally-Coupled Oscillators   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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The categories of this paper are: cs
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: chem-ph
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The categories of this paper are: physics
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph