Abstract
Many real-world systems can be modeled as networks of interacting oscillatory units. Collective dynamics that are of functional relevance for the oscillator network, such as switching between metastable states, arise through the interplay of network structure and interaction. Here, we give results for small networks on the existence of heteroclinic cycles between dynamically invariant sets on which the oscillators show localized frequency synchrony. Trajectories near these heteroclinic cycles will exhibit sequential switching of localized frequency synchrony: a population oscillators in the network will oscillate faster (or slower) than others and which population has this property sequentially changes over time. Since we give explicit conditions on the system parameters for such dynamics to arise, our results give insights into how network structure and interactions (which include higher-order interactions between oscillators) facilitate heteroclinic switching between localized frequency synchrony.
Highlights
Networks of interacting oscillatory units can give rise to dynamics where the system appears to be in one metastable state before “switching” to another in a rapid transition
The heteroclinic cycles lead to switching between localized frequency synchrony which are observed in numerical simulations
Phase oscillator networks with higher-order interactions can give rise to heteroclinic cycles between frequency synchrony; in numerical simulations these lead to sequential acceleration and deceleration of oscillator populations
Summary
Networks of interacting oscillatory units can give rise to dynamics where the system appears to be in one metastable state before “switching” to another in a rapid transition. Even networks of identical phase oscillators that are organized into different populations can give rise to dynamics where frequency synchrony is local to a population rather than global across the whole network. In contrast to attracting sets with localized frequency synchrony, the dynamics here induce sequential switching dynamics: Which population of oscillators oscillates at a faster (or slower) rate will change over time These results are of interest from several distinct perspectives. They illuminate how the interplay of network structure and functional interactions between units gives rise to heteroclinic dynamics in phase oscillator networks: We explicitly relate the network coupling parameters to the existence of heteroclinic cycles.
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