Solitary states and partial synchrony in oscillatory ensembles with attractive and repulsive interactions

Abstract

We numerically and analytically analyze transitions between different synchronous states in a network of globally coupled phase oscillators with attractive and repulsive interactions. The elements within the attractive or repulsive group are identical, but natural frequencies of the groups differ. In addition to a synchronous two-cluster state, the system exhibits a solitary state, when a single oscillator leaves the cluster of repulsive elements, as well as partially synchronous quasiperiodic dynamics. We demonstrate how the transitions between these states occur when the repulsion starts to prevail over attraction.PACS numbers: 05.45.Xt Synchronization; coupled oscillators Networks of coupled oscillators are a popular model for many engineered or natural systems. The main effect -emergence of a collective mode via synchronization -is now well-understood and therefore focus of research shifted recently to analysis of different complex states. These states include chimeras, when a population of identical units splits into a synchronous and asynchronous part, quasiperiodic partially synchronous states, characterized by the difference of frequencies of individual units and of the collective mode, and clusters and heteroclinic cycles, to name just a few. Of particular interest are ensembles where some elements have only attractive connections while others have only repulsive ones. This model is motivated by studies of neuronal networks that are built from excitatory and inhibitory neurons. In this paper we analyze how the state of such a setup changes with the interplay of attraction and repulsion. We demonstrate that if the frequency mismatch between attractive and repulsive units is smaller than some critical value then desynchronization occurs via appearance of the solitary state. With the further increase of repulsion the system undergoes a transition to quasiperiodic partial synchrony. In the latter state the attractive units remain synchronized, while the repulsive group settles between synchrony and asynchrony so that the mean fields of both groups remain locked, but the frequency of the repulsive elements is larger than that of their mean field. For a large frequency mismatch of attractive and repulsive groups desynchronization immediately leads to partial synchrony.

Solitary states in adaptive nonlocal oscillator networks

“…The stable equilibria at θ = 0 and θ = π correspond to in-phase synchronous and antipodal where a 1 = π and a i =1 = 0, respectively. The other two saddle equilibria correspond to the special class of double antipodal states [36] and describe therefore phase-cluster similar to those described in [50,72]. While these equilibria can be stable for the reduced system (21)- (22), they are always unstable for (1)- (2) in case of global coupling [37].…”

“…The solitary states are particular examples of multiclusters with a large group of frequency synchronized oscillators (background cluster) and individual solitary nodes with different frequency, i.e., clusters consisting of only one oscillator. These special kind of states, for which we provide an analysis of their emergence in Section 5, are of particular interest as they are found in various dynamical systems [48,50,53,54,61,[66][67][68][69].…”

“…A particular case of hierarchical multicluster states are solitary states for which only one single element behaves differently compared with the behavior of the background group, i.e., the neighboring elements. These states have been found in diverse dynamical systems such as generalized Kuramoto-Sakaguchi models [48][49][50][51], the Kuramoto model with inertia [52][53][54], the Stuart-Landau model [55], the FitzHugh-Nagumo model [56,57], systems of excitable units [58] as well as Lozi maps [59] and even experimental setups of coupled pendula [60]. Solitary states are considered as important states in the transition from coherent to incoherent dynamics [52,61,62].…”

“…Despite their appearance in many well-studied models, the mechanisms of their emergence are less understood. Until now, only a few works could shed some light on the details behind their formation [48,50,53,63].…”

Machine Learning Assisted Chimera and Solitary States in Networks

“…Hence, k-solitary states comprise k isolated elements [25]. Recently, the existence of solitary states has been demonstrated in a network of ensembles having attractive and repulsive interactions at the edge of synchrony [26] and partial synchrony [27], inertial Kuramoto model [28], oscillators with negative time-delayed feedback under external forcing [29], identical populations of Stuart-Landau oscillators [30], FitzHugh-Nagumo neurons in the oscillatory regime [4], and neuronal oscillators and coupled chaotic maps in the presence of delayed links [31]. The occurrence of solitary states can be observed in power grid networks in which individual gridunits gradually desynchronize during a partial or complete blackout [32].…”