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

Abstract

Abstract-We present new necessary and sufficient conditions for the existence of fixed points in a finite system of coupled phase oscillators on a complete graph. We use these conditions to derive bounds on the critical coupling.

There is no non-zero stable fixed point for dense networks in the homogeneous Kuramoto model

“…This is the zero fixed point solution defined by all phase angle differences being zero. This result together with existing research proves a conjecture of Verwoerd and Mason (2007) that for the complete network and homogeneous model the zero fixed point has a basin of attraction consisting of the entire space minus a set of measure zero. The necessary conditions are also tested to see how close to sufficiency they might be by applying them to a class of regular degree networks studied by Wiley, Strogatz and Girvan (2006).…”

“…Thus the characterization of the stable fixed points in the homogeneous case is a valuable place to start in understanding the general (inhomogeneous) case. Despite this, and the interest in understanding this problem (see Jadbabaie, Motee, and Barahona (2004), Wiley, Strogatz and Girvan (2006), Ochab andGora (2009), andVerwoerd andMason (2007)), there is a dearth of results concerning stable fixed points over the full space of phase angles [−,], specifically the conditions on network topology for non-zero stable fixed points.…”

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

“…In this paper, we shall be concerned with synchronization in finite systems of coupled oscillators. Specifically: we shall establish (new) necessary and sufficient conditions for the existence of fixed points in a finite system of coupled oscillators (see also [25,26]); compute bounds on the critical coupling strength for such systems; and provide insights into the number of fixed points possible under strong coupling. Our analysis is in the spirit of the work presented in [9,3], and places particular emphasis on the existence of fixed points.…”

Exotic equilibria of Harary graphs and a new minimum degree lower bound for synchronization

“…However it seems that they were not aware of each other and some repetition on the results arised. For instance, as far as 2006, in [11], it is proved that the complete graphs synchronize, but the question is conjectured two years later in [15] and proved (again) in [14]. Later on, we made some improvements: we proved that the AGS property depends only in the block of the graphs [3], we also proved that every connected graph is the induced graph of a synchronized one, and that any graph with at least one cycle is homeomorphic to a non synchronized one [4].…”