Oscillators as Systems and Synchrony as a Design Principle

Abstract

Summary.The chapter presents an expository survey of ongoing research by the author on a system theory for oscillators. Oscillators are regarded as open systems that can be interconnected to robustly stabilize ensemble phenomena characterized by a certain level of synchrony. The first part of the chapter provides examples of design (stabilization) problems in which synchrony plays an important role. The second part of the chapter shows that dissipativity theory provides an interconnection theory for oscillators.

Global Phase-Locking in Finite Populations of Phase-Coupled Oscillators

“…We have that g(0) = 0 and det( Lastly, note that continuity of g −1 implies that the fixed points of the original system (17) will converge to the fixed points of the homogeneous system (21) as k tends to infinity.…”

Conditions for the Existence of Fixed Points in a Finite System of Kuramoto Oscillators

“…INTRODUCTION Synchronized behaviour has been widely observed in natural and engineered systems [13], [2], [12], [5], and understanding the mechanisms behind its emergence is a key issue in the study of interconnected dynamical systems. For some time now, there has been considerable interest across the mathematics, physics and engineering communities in the development and analysis of simple mathematical models of synchronization [8], [14], [15], [3], [16], [17]. One of the most popular frameworks for the mathematical study of synchronization is the so-called Kuramoto model of phase coupled oscillators [9], [10], [18], [19], [1].…”

Kick synchronization versus diffusive synchronization

“…This section provides a short introduction to oscillators viewed as open dynamical systems, that is, as dynamical systems that interact with their environment [67]. We first recall basic definitions about stable periodic orbits in n-dimensional state-space models (see [21,28] for details).…”