Metastability in the stochastic nearest-neighbour Kuramoto model of coupled phase oscillators

Abstract The Kuramoto model (KM) of n coupled phase-oscillators is analysed in this work. The KM on a Cayley graph possesses a family of steady state solutions called twisted states. Topologically distinct twisted states are distinguished by the winding number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> </mml:math> . It is known that for the KM on the nearest-neighbour graph, a q -twisted state is stable if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mi>q</mml:mi> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> . In the presence of small noise, the KM exhibits metastable transitions between q –twisted states. Specifically, a typical trajectory remains in the basin of attraction of a given q -twisted state for an exponentially long time, but eventually transitions to the vicinity of another such state. In the course of this transition, it passes in close proximity of a saddle of Morse index 1, called a relevant saddle. In this work, we provide an exhaustive analysis of metastable transitions in the stochastic KM with nearest-neighbour coupling. We start by analysing the equilibria and their stability. First, we identify all equilibria in this model. Using the discrete Fourier transform and eigenvalue estimates for rank–1 perturbations of symmetric matrices, we classify the equilibria by their Morse indices. In particular, we identify all stable equilibria and all relevant saddles involved in the metastable transitions. Further, we use Freidlin–Wentzell theory and the potential-theoretic approach to metastability to establish the metastable hierarchy and sharp estimates of Eyring–Kramers type for the transition times. The former determines the precise order, in which the metastable transitions occur, while the latter characterises the times between successive transitions. The theoretical estimates are complemented by numerical simulations and a careful numerical verification of the transition times. Finally, we discuss the implications of this work for the KM with other coupling types including non-local coupling and the continuum limit as n tends to infinity.