A minimal model of self-consistent partial synchrony

Abstract

In the body of the paper we have spotted three minor errors that do not affect the overall results and have to be corrected as follows:(A) Z m as defined in equations (10) and (12) coincide with the usual Kuramoto-Daido order parameters (see equation (3)), only once they have been multiplied by 2p. All formulas concerning the stability of SCPS remain unchanged, while the expression of the exponent 1 d , controlling the stability of the splay state (see equation (6)) must be divided by 2p, i.e. AbstractWe show that self-consistent partial synchrony in globally coupled oscillatory ensembles is a general phenomenon. We analyze in detail appearance and stability properties of this state in possibly the simplest setup of a biharmonic Kuramoto-Daido phase model as well as demonstrate the effect in limit-cycle relaxational Rayleigh oscillators. Such a regime extends the notion of splay state from a uniform distribution of phases to an oscillating one. Suitable collective observables such as the Kuramoto order parameter allow detecting the presence of an inhomogeneous distribution. The characteristic and most peculiar property of self-consistent partial synchrony is the difference between the frequency of single units and that of the macroscopic field.

Dynamics and stability of chimera states in two coupled populations of oscillators

“…Such a state is similar to that of self-consistent partial synchrony [22][23][24] Regarding the types of oscillators used, early works used phase oscillators with sinusoidal interaction functions [4,5], while later studies include oscillators near a SNIC bifurcation [25], van der Pol oscillators [26], oscillators with inertia [27][28][29], Stuart-Landau oscillators [15,30], and neuron models including leaky integrate-and-fire [31], quadratic integrate-and-fire [32], and FitzHugh-Nagumo [3].…”

Noise-induced stabilization of collective dynamics

“…In the previous section we have, however, seen that even a very small noise is sufficient to stabilize the It is therefore natural to ask whether this scenario is peculiar of the biharmonic setup. We check this point by studying another model, an ensemble of mean-field coupled Rayleigh oscillators, where SCPS has been observed and found to lose stability in a purely deterministic setup [6]. We show that a small amount of noise is again able to stabilize SCPS.…”

“…Here, we approach the problem from the macroscopic point of view, extending the method introduced in Ref. [6] to account for the presence of microscopic noise.…”

“…Therefore, the determination of P 0 requires finding a fixed point in a four-dimensional space: this problem was tackled and solved numerically in Ref. [6].…”

“…In this paper we discuss another such instance, where a finite but small amount of white noise, acting independently on an ensemble of identical oscillators, stabilizes self-consistent partial synchrony (SCPS), an ubiquitous, collective regime [6] observed in ensembles of identical oscillators. This phenomenon differs from standard noise-induced bifurcations (see, e.g.…”