Between phase and amplitude oscillators

Abstract

We analyze an intermediate collective regime where amplitude oscillators distribute themselves along a closed, smooth, time-dependent curve C, thereby maintaining the typical ordering of (identical) phase oscillators. This is achieved by developing a general formalism based on two partial differential equations, which describe the evolution of the probability density along C and of the shape of C itself. The formalism is specifically developed for Stuart-Landau oscillators, but it is general enough to be applied to other classes of amplitude oscillators. The main achievements consist in: (i) identification and characterization of a new transition to selfconsistent partial synchrony (SCPS), which confirms the crucial role played by higher Fourier hamonics in the coupling function; (ii) an analytical treatment of SCPS, including a detailed stability analysis; (iii) the discovery of a new form of collective chaos, which can be seen as a generalization of SCPS and characterized by a multifractal probability density. Finally, we are able to describe given dynamical regimes both at the macroscopic as well as the microscopic level, thereby shedding further light on the relationship between the two different levels of description.

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

“…gives −2/ = −40, for this solution. The second group of eigenvalues is presumably related to the dynamics of P , and has been observed in other similar systems [2,22,24]. The slight deviation from the imaginary axis visible in panel (c) of Fig.…”

“…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].…”

“…We have used the results of [22] to study the dynamics of chimera states in networks formed from two populations of identical oscillators, with different strengths of coupling both within and between populations. We studied four different types of oscillators.…”

“…In this paper we use techniques from [22] to revisit the system studied in [30] and calculate stability information for the solutions found there. Since the approach in [22] is generally applicable to a situation in which oscillators in one population lie on a closed smooth curve, we then apply these ideas to three more networks formed from two coupled populations. The second network we consider consists of Kuramoto oscillators with inertia, each described by a second order ODE.…”

Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation

“…Therefore, it is assumed hereafter that the term NUIS is constrained to f n = 0 (n > 2). [24,36]. We determined the boundary through simulations with N = 200 oscillators, but the result is insensitive if a larger N value is used.…”