Reduced dynamics and symmetric solutions for globally coupled weakly dissipative oscillators
Reduced dynamics and symmetric solutions for globally coupled weakly dissipative oscillators
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全局耦合弱耗散振荡器的简化动力学和对称解
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
G. Dangelmayr
中科院分区：
文献类型：
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作者：
Peter Ashwin;G. Dangelmayr
耦合振荡器系统可能会表现出吸引对称性破缺的同步簇的自发动态形成；这有助于模拟各种物理过程。对于任意非线性振荡器，同步簇状态稳定性的分析计算通常是不可能的。在本文中，我们研究了一类特殊的强非线性振荡器，它们在分析上易于处理。我们研究了周期性振荡器的等时性（周期对能量依赖性的转折点）对全局耦合振荡器网络的集群状态的影响。我们扩展了先前关于弱耗散全局耦合非线性哈密顿振子网络的工作，在耗散和耦合很小且阶数相似的假设下，给出了某些簇周期态的存在和稳定性的条件。这通过对振子示例系统的数值模拟得到了验证，该振子系统是具有四次势的平面哈密顿振子的弱耗散扰动。最后，我们使用从弱耗散情况导出的简化相能量模型来激发一类新的相能量模型，该模型可有效用于理解更一般的耦合振荡器系统中的聚类和环面破裂等效应。我们发现等时性的性质可以有效地推广到此类系统，并且我们研究了它们吸引动态的一些例子。
Systems of coupled oscillators may exhibit spontaneous dynamical formation of attracting synchronized clusters with broken symmetry; this can be helpful in modelling various physical processes. Analytical computation of the stability of synchronized cluster states is usually impossible for arbitrary nonlinear oscillators. In this paper we examine a particular class of strongly nonlinear oscillators that are analytically tractable. We examine the effect of isochronicity (a turning point in the dependence of period on energy) of periodic oscillators on clustered states of globally coupled oscillator networks. We extend previous work on networks of weakly dissipative globally coupled nonlinear Hamiltonian oscillators to give conditions for the existence and stability of certain clustered periodic states under the assumption that dissipation and coupling are small and of similar order. This is verified by numerical simulations on an example system of oscillators that are weakly dissipative perturbations of a planar Hamiltonian oscillator with a quartic potential. Finally we use the reduced phase-energy model derived from the weakly dissipative case to motivate a new class of phase-energy models that can be usefully employed for understanding effects such as clustering and torus breakup in more general coupled oscillator systems. We see that the property of isochronicity usefully generalizes to such systems, and we examine some examples of their attracting dynamics.