Chaos in generically coupled phase oscillator networks with nonpairwise interactions

Abstract

The Kuramoto-Sakaguchi system of coupled phase oscillators, where interaction between oscillators is determined by a single harmonic of phase differences of pairs of oscillators, has very simple emergent dynamics in the case of identical oscillators that are globally coupled: there is a variational structure that means the only attractors are full synchrony (in-phase) or splay phase (rotating wave/full asynchrony) oscillations and the bifurcation between these states is highly degenerate. Here we show that nonpairwise coupling-including three and four-way interactions of the oscillator phases-that appears generically at the next order in normal-form based calculations, can give rise to complex emergent dynamics in symmetric phase oscillator networks. In particular, we show that chaos can appear in the smallest possible dimension of four coupled phase oscillators for a range of parameter values.In this paper, we show that symmetrically coupled phase oscillators with generic (but nonpairwise) interactions yield rich dynamics even for as few as N = 4 oscillators. Although the lowest order approximation of a phase reduction of symmetrically coupled oscillators close to a Hopf bifurcation has only the Kuramoto-Sakaguchi first harmonic interaction terms, the next order includes generic terms with both second harmonic pairwise interactions and interactions of up to four phases 9 . The symmetries we consider imply the existence of invariant subspaces such that the ordering of phases is preserved 10 . In contrast to the Kuramoto-Sakaguchi equations, the additional nonpairwise interaction terms mean we can find attracting chaos for a range of normal form parameter values. As a consequence, the phase dynamics of generic weakly coupled oscillators will be quite rich and chaos can occur even for the phase dynamics in the weak coupling limit without amplitude degrees of freedom 30 .

Heteroclinic switching between chimeras

“…A full analysis of this system (and its nonpairwise approximation) is beyond the scope of this article. Discussion-Phase oscillator networks with nonpairwise coupling have surprisingly rich dynamics [19,[22][23][24]; here, nonpairwise interaction allows to show the existence of heteroclinic connections between weak chimeras. Here nonpairwise coupling arises through a bifurcation parameter that depends on local order parameters of different populations.…”

Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplexes

“…Xt,89.75.Hc Research into the macroscopic dynamics of large ensembles of coupled oscillators have extended our understanding of natural and engineered systems ranging from cell cycles to power grids [1][2][3][4][5]. However, with few exceptions (including [6][7][8]), little attention has been paid to the synchronization dynamics of coupled oscillator systems where interactions are not pair-wise, but rather n-way, with n ≥ 3. Such interactions are called "simplicial", where an n-simplex represents an interaction between n + 1 units, so 2-simplices describe three-way interactions, etc [9].…”

Reconstruction of a random phase dynamics network from observations

“…Recently, more general cases of phase coupling have been considered [9]. The next level of complexity in comparison to the pairwise coupling networks (4) are the hypernetworks with triple coupling terms [21,22,23]…”