Optimal cost tuning of frustration: Achieving desired states in the Kuramoto-Sakaguchi model

TL;DR

This paper develops a formalism to compute optimal phase lag parameters in the Kuramoto-Sakaguchi model, enabling desired phase configurations and improving synchronization, with potential applications in complex networked oscillatory systems.

Contribution

It introduces a method to determine phase lag parameters that achieve specific phase states in the Kuramoto-Sakaguchi model, including a cost minimization approach.

Findings

Optimal parameters enhance frequency synchronization in nonlinear models.

A continuous spectrum of solutions exists for phase configurations.

The formalism applies to full synchronization and symmetric states.

Abstract

There are numerous examples of studied real-world systems that can be described as dynamical systems characterized by individual phases and coupled in a network like structure. Within the framework of oscillatory models, much attention has been devoted to the Kuramoto model, which considers a collection of oscillators interacting through a sinus function of the phase differences. In this paper, we draw on an extension of the Kuramoto model, called the Kuramoto-Sakaguchi model, which adds a phase lag parameter to each node. We construct a general formalism that allows to compute the set of lag parameters that may lead to any phase configuration within a linear approximation. In particular, we devote special attention to the cases of full synchronization and symmetric configurations. We show that the set of natural frequencies, phase lag parameters and phases at the steady state is coupled by an equation and a continuous spectra of solutions is feasible. In order to quantify the system's strain to achieve that particular configuration, we define a cost function and compute the optimal set of parameters that minimizes it. Despite considering a linear approximation of the model, we show that the obtained tuned parameters for the case of full synchronization enhance frequency synchronization in the nonlinear model as well.