具有时滞和分布剪切的Kuramoto模型的多参数分岔分析
Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear
摘要：
本文在仓本模型中考虑了时滞效应和分布剪力。通过对约化泛函微分方程的特征方程的分析，在N-Antonsen流形上得到了非相干态在多参数空间中的稳定性边界。此外，非常丰富的动力学行为，如稳定开关诱导同步开关可以出现在这个方程。当系统失去稳定性时，系统出现Hopf分岔相干态，利用规范形理论和中心流形定理确定了系统发生Hopf分岔的临界性。一方面，理论分析表明剪切分布宽度和时间延迟都可以消除同步，导致仓本模型的非相干。另一方面，时间延迟可以诱导多个相干态共存。最后，给出了一些数值模拟来支持所得到的结果，其中几个分支图绘制，并讨论了时间延迟和剪切的影响。
英文摘要：
In this paper, time delay effect and distributed shear are considered in the Kuramoto model. On the Ott-Antonsen's manifold, through analyzing the associated characteristic equation of the reduced functional differential equation, the stability boundary of the incoherent state is derived in multiple-parameter space. Moreover, very rich dynamical behavior such as stability switches inducing synchronization switches can occur in this equation. With the loss of stability, Hopf bifurcating coherent states arise, and the criticality of Hopf bifurcations is determined by applying the normal form theory and the center manifold theorem. On one hand, theoretical analysis indicates that the width of shear distribution and time delay can both eliminate the synchronization then lead the Kuramoto model to incoherence. On the other, time delay can induce several coexisting coherent states. Finally, some numerical simulations are given to support the obtained results where several bifurcation diagrams are drawn, and the effect of time delay and shear is discussed.
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