Almost global convergence to practical synchronization in the generalized Kuramoto model on networks over the n-sphere

Johan Markdahl,Daniele Proverbio,La Mi,Jorge Goncalves

doi:10.1038/s42005-021-00689-y

Johan Markdahl, Daniele Proverbio + Show 2 more

Open Access

https://doi.org/10.1038/s42005-021-00689-y

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Journal: Communications Physics	Publication Date: Aug 20, 2021
Citations: 12	License type: open-access

Affiliation: University of Luxembourg

Abstract

From the flashing of fireflies to autonomous robot swarms, synchronization phenomena are ubiquitous in nature and technology. They are commonly described by the Kuramoto model that, in this paper, we generalise to networks over n-dimensional spheres. We show that, for almost all initial conditions, the sphere model converges to a set with small diameter if the model parameters satisfy a given bound. Moreover, for even n, a special case of the generalized model can achieve phase synchronization with nonidentical frequency parameters. These results contrast with the standard n = 1 Kuramoto model, which is multistable (i.e., has multiple equilibria), and converges to phase synchronization only if the frequency parameters are identical. Hence, this paper shows that the generalized network Kuramoto models for n ≥ 2 displays more coherent and predictable behavior than the standard n = 1 model, a desirable property both in flocks of animals and for robot control.

Highlights

From the flashing of fireflies to autonomous robot swarms, synchronization phenomena are ubiquitous in nature and technology
Other topics of interest include characterizing the equilibria of the homogeneous network Kuramoto model and their basin of attraction by theoretical and numerical means[23,24]
Our model extends the ndimensional sphere Kuramoto model considered by Chandra, Girvan, and Ott[39,40,41] to complex networks on ellipsoids

Summary

Introduction

From the flashing of fireflies to autonomous robot swarms, synchronization phenomena are ubiquitous in nature and technology. Other topics of interest include characterizing the equilibria of the homogeneous network Kuramoto model (i.e., with identical natural frequencies) and their basin of attraction by theoretical and numerical means[23,24]. For n ≥ 2, i.e., for all dimensions except that of the standard Kuramoto model (n = 1 in our notation), we find that the systems asymptotically converges to synchronization almost globally (i.e., from almost all initial conditions), provided that the spread of the frequency matrices is sufficiently small in the norm.

Results

Conclusion