THE DYNAMICS OF N-WEAKLY COUPLED IDENTICAL OSCILLATORS
THE DYNAMICS OF N-WEAKLY COUPLED IDENTICAL OSCILLATORS
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DOI：
10.1007/bf02429852
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发表时间：
1992-01-01
影响因子：
3
通讯作者：
SWIFT, JW
中科院分区：
文献类型：
--
作者：
ASHWIN, P;SWIFT, JW
我们提出了一个框架，分析任意网络的相同耗散振荡器假设弱耦合。使用网络的对称性，我们发现在相空间中存在的动态不变的区域纯粹凭借其时空对称性（时间对称性对应于相移）。我们专注于阵列是对称的振荡器的所有排列下（这与全球耦合），也对环的振荡器与定向和双向耦合。对于这些例子，我们分类所有的时空对称性，包括极限环的解决方案，如同相振荡和那些涉及相移。在一般条件下，我们还证明了“次极大”极限环解的存在性。定义了相空间的正则不变区域，并利用它研究了系统的动力学行为。我们讨论了极限环如何失去和获得稳定性，以及对称性如何产生结构稳定的异宿环，这种现象一般不会在没有对称性的系统中发现。我们还研究了某些类型的耦合（包括具有对称波形的振荡器之间的线性耦合）如何引起退化行为，其中振荡器解耦成更小的组。
We present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling. Using the symmetry of the network, we find dynamically invariant regions in the phase space existing purely by virtue of their spatio-temporal symmetry (the temporal symmetry corresponds to phase shifts). We focus on arrays which are symmetric under all permutations of the oscillators (this arises with global coupling) and also on rings of oscillators with both directed and bidirectional coupling. For these examples, we classify all spatio-temporal symmetries, including limit cycle solutions such as in-phase oscillation and those involving phase shifts. We also show the existence of "submaximal" limit cycle solutions under generic conditions. The canonical invariant region of the phase space is defined and used to investigate the dynamics. We discuss how the limit cycles lose and gain stability, and how symmetry can give rise to structurally stable heteroclinic cycles, a phenomenon not generically found in systems without symmetry. We also investigate how certain types of coupling (including linear coupling between oscillators with symmetric waveforms) can give rise to degenerate behaviour, where the oscillators decouple into smaller groups.