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We explore chaos in the Kuramoto model with multimodal distributions of thenatural frequencies of oscillators and provide a comprehensive descriptionunder what conditions chaos occurs. For a natural frequency distribution with$M$ peaks it is typical that there is a range of coupling strengths such thatoscillators belonging to each peak form a synchronized cluster, but theclusters do not globally synchronize. We use collective coordinates to describethe inter- and intra-cluster dynamics, which reduces the Kuramoto model to$2M-1$ degrees of freedom. We show that under some assumptions, there is atime-scale splitting between the slow intracluster dynamics and fastintercluster dynamics, which reduces the collective coordinate model to an$M-1$ degree of freedom rescaled Kuramoto model. Therefore, four or moreclusters are required to yield the three degrees of freedom necessary forchaos. However, the time-scale splitting breaks down if a clusterintermittently desynchronizes. We show that this intermittent desynchronizationprovides a mechanism for chaos for trimodal natural frequency distributions. Inaddition, we use collective coordinates to show analytically that chaos cannotoccur for bimodal frequency distributions, even if they are asymmetric and ifintermittent desynchronization occurs.

Chaos in networks of coupled oscillators with multimodal natural frequency distributions