AMPLITUDE EXPANSIONS FOR INSTABILITIES IN POPULATIONS OF GLOBALLY-COUPLED OSCILLATORS
AMPLITUDE EXPANSIONS FOR INSTABILITIES IN POPULATIONS OF GLOBALLY-COUPLED OSCILLATORS
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DOI：
10.1007/bf02188217
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发表时间：
1994-03-01
影响因子：
1.6
通讯作者：
CRAWFORD, JD
中科院分区：
文献类型：
--
作者：
CRAWFORD, JD
本文分析了耦合振子平均场模型中非相干态附近的非线性动力学。 粒子布居由Fokker-Planck方程描述相位分布，我们应用中心流形约化得到了从具有均匀相位分布的平衡态出发的稳态和Hopf分岔的振幅方程. 当人口是由一个固有的频率分布是反射对称的零，这个问题有圆对称。 在零外部噪声的限制下，尽管临界特征值被嵌入在连续谱中，但振幅方程中的非线性系数仍然是有限的，与Vlasov-Poisson方程描述的类似不稳定性中的奇异行为相反。 对于双峰反射对称分布，这两种类型的分歧是可能的，他们在一个余维2 Takens-Bogdanov点重合。 稳态分叉可以是超临界或亚临界的，并产生与时间无关的同步状态。 Hopf分岔产生超临界稳定驻波和超临界不稳定行波。 以前的工作在一个双峰人口的霍普夫分岔的Bonilla，Neu和Spigler和奥田和仓本预测稳定的行波和稳定的驻波，分别。 这些以前的计算结果的比较表明，稳定的行波的预测结果从失败，包括所有不稳定的模式。
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 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.