The Kuramoto model in presence of additional interactions that break rotational symmetry

TL;DR

This paper explores a generalized Kuramoto model with symmetry-breaking interactions, revealing a complex phase diagram with stationary and standing wave phases, supported by numerical and analytical methods including the Ott-Antonsen ansatz.

Contribution

It introduces a symmetry-breaking term into the Kuramoto model and analyzes its effects, uncovering new stationary and oscillatory phases not present in the original model.

Findings

Existence of stationary states in the laboratory frame due to symmetry breaking.

Identification of standing wave phases with oscillatory order parameters.

Development of a reduced dynamical system using the Ott-Antonsen ansatz.

Abstract

The Kuramoto model serves as a paradigm to study the phenomenon of spontaneous collective synchronization. We study here a nontrivial generalization of the Kuramoto model by including an interaction that breaks explicitly the rotational symmetry of the model. In an inertial frame (e.g., the laboratory frame), the Kuramoto model does not allow for a stationary state, that is, a state with time-independent value of the so-called Kuramoto (complex) synchronization order parameter $z\equiv re^{i\psi}$; Note that a time-independent $z$ implies $r$ and $\psi$ both time independent, with the latter fact corresponding to a state in which $\psi$ rotates at zero frequency (no rotation). In this backdrop, we ask: Does the introduction of the symmetry-breaking term suffice to allow for the existence of a stationary state in the laboratory frame? Compared to the original model, we reveal a rather rich phase diagram of the resulting model, with the existence of both stationary and standing wave phases. While in the former the synchronization order parameter $r$ has a long-time value that is time independent, one has in the latter an oscillatory behavior of the order parameter as a function of time that nevertheless yields a non-zero and time-independent time average. Our results are based on numerical integration of the dynamical equations as well as an exact analysis of the dynamics by invoking the so-called Ott-Antonsen ansatz that allows to derive a reduced set of time-evolution equations for the order parameter.