Almost Global Synchronization of Symmetric Kuramoto Coupled Oscillators

Abstract

A few decades ago, Y. Kuramoto introduced a mathematical model of weakly coupled oscillators that gave a formal framework to some of the works of A.T. Winfree on biological clocks [Kuramoto (1975), Kuramoto (1984), Winfree (1980)]. The model proposes the idea that several oscillators can interact in a way such that the individual oscillation properties change in order to achieve a global behavior for the interconnected system. The Kuramoto model serves as a good representation of many systems in several contexts: biology, engineering, physics, mechanics, etc. [Ermentrout (1985), York (1993), Strogatz (1994), Dussopt et al. (1999), Strogatz (2000), Jadbabaie et a. (2003), Rogge et al. (2004), Marshall et al. (2004), Moshtagh et al. (2005)].
Recently, many works on the control community have focused on the analysis of the Kuramoto model, specially the one with sinusoidal coupling. The consensus or collective synchronization of the individuals is particularly important in many applications representing coordination, cooperation, emerging behavior, etc. Local stability properties of the consensus have been initially explored in [Jadbabaie et al. (2004)]. It must be noted that little attention has been devoted to the influence of the underlying interconnection graph on the stability properties of the system. The reason could be the fact that the local stability does not depend on the interconnection [van Hemmen et al. (1993)]. Global or almost global dynamical properties were studied in [Monzón et al. (2005), Monzón (2006), Monzón et al.  (2006)]. In these works, the relevance of the interconnection graph of the system was hinted.  In the present chapter, we go deeper on the analysis of the relationships between the dynamical properties of the system and the algebraic properties of the interconnection graph, exploiting the strong algebraic structure that every graph has. We step forward into a classification of the interconnection graphs that ensure almost global attraction of the set of synchronized states.
In Section 2 we present the Kuramoto model for sinusoidally coupled oscillators, its general properties and the notion of almost global synchronization; in Section 3 we review some basic facts on algebraic graph theory; the symmetric Kuramoto model and the block analysis are presented in Sections 4 and 5; Section 6 gives some examples and applications of the main results; Section 7 presents the problem of classification of almost global synchronizing topologies.