From phase to amplitude oscillators

TL;DR

This paper develops a formalism to analyze collective dynamics of amplitude oscillators, revealing new transitions to self-consistent partial synchrony, stability properties, and a novel form of collective chaos with multifractal characteristics.

Contribution

It introduces a general PDE-based framework for amplitude oscillators, identifies a new transition to SCPS, and uncovers a new type of collective chaos.

Findings

Identification of a new transition to self-consistent partial synchrony.

Analytical stability analysis of SCPS.

Discovery of a new form of collective chaos with multifractal density.

Abstract

We analyze an intermediate collective regime where amplitude oscillators distribute themselves along a closed, smooth, time-dependent curve $\mathcal{C}$, 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 $\mathcal{C}$ and of the shape of $\mathcal{C}$ 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 self-consistent 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.