Heteroclinic Dynamics of Localized Frequency Synchrony: Heteroclinic Cycles for Small Populations

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 changes sequentially 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.

Almost Complete and Equable Heteroclinic Networks

“…4.1. Other authors have implicitly noted the importance of graph structures such as ∆-cliques for properties of clean heteroclinic networks [7,13]. Note that a clean network need not be equable: we give an example for this in Subsection 4.2.…”

“…It is also of potential interest in applications such as design of computational systems that permit only certain transitions. Several recent papers, Ashwin and Postlethwaite [4,5], Bick [7] and Field [12,13], have considered several approaches to the design of systems that have specific heteroclinic networks. These approaches to the realization of a graph as a heteroclinic network typically result in networks that are not asymptotically stable or even contain unstable manifolds of all saddles.…”

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

“…Consequently, we apply the stability results of Garrido-da-Silva and Castro (2019) to calculate the stability indices. We first consider three coupled oscillator populations to calculate stability indices for the heteroclinic cycles in Bick (2019). We then show that four coupled oscillator populations support a heteroclinic network which contains two distinct heteroclinic cycles of the type considered before.…”

“…Heteroclinic dynamics are not only of interest in their own right (see Weinberger and Ashwin 2018 for a recent review), but have also been related, for example, to computation in neural systems (Rabinovich et al 2006;Neves and Timme 2012;). In the companion paper (Bick 2019), we showed the existence of heteroclinic cycles between invariant sets with localized frequency synchrony in three coupled populations. In particular, we obtained explicit existence conditions for heteroclinic cycles in terms of the interaction between the phase oscillator populations.…”

“…These synchrony patterns include patterns of localized frequency synchrony where some populations oscillate faster (or slower) than others (Ashwin and Burylko 2015;Bick and Ashwin 2016;Omel'chenko 2018). For coupled oscillator populations with higher-order interactions-including higher harmonics or nonlinear interactions between three or more oscillators (nonpairwise terms)-which arise naturally in phase reductions (Ashwin and Rodrigues 2016;León and Pazó 2019), heteroclinic structures between patterns of localized frequency synchrony are possible (Bick 2018(Bick , 2019. Heteroclinic dynamics are not only of interest in their own right (see Weinberger and Ashwin 2018 for a recent review), but have also been related, for example, to computation in neural systems (Rabinovich et al 2006;Neves and Timme 2012;).…”