Heteroclinic Dynamics of Localized Frequency Synchrony: Stability of Heteroclinic Cycles and Networks

Abstract

In the first part of this paper, we showed that three coupled populations of identical phase oscillators give rise to heteroclinic cycles between invariant sets where populations show distinct frequencies. Here, we now give explicit stability results for these heteroclinic cycles for populations consisting of two oscillators each. In systems with four coupled phase oscillator populations, different heteroclinic cycles can form a heteroclinic network. While such networks cannot be asymptotically stable, the local attraction properties of each cycle in the network can be quantified by stability indices. We calculate these stability indices in terms of the coupling parameters between oscillator populations. Hence, our results elucidate how oscillator coupling influences sequential transitions along a heteroclinic network where individual oscillator populations switch sequentially between a high and a low frequency regime; such dynamics appear relevant for the functionality of neural oscillators.

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Cited by in corpus (10)

• What are higher-order networks?
• Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes
• Multi-Population Phase Oscillator Networks with Higher-Order Interactions
• Heteroclinic Dynamics of Localized Frequency Synchrony: Heteroclinic Cycles for Small Populations
• Collective dynamics on higher-order networks
• Chaos in Coupled Heteroclinic Cycles and its Piecewise-Constant Representation
• Heteroclinic Switching between Chimeras in a Ring of Six Oscillator Populations
• Heteroclinic Dynamics in Network Dynamical Systems with Higher-Order Interactions
• Heteroclinic cycles and chaos in a system of four identical phase oscillators with global biharmonic coupling
• How many points converge to a heteroclinic network in an aperiodic way?