Codimension-two bifurcations induce hysteresis behavior and multistabilities in delay-coupled Kuramoto oscillators

Abstract

Hysteresis phenomena and multistability play crucial roles in the dynamics of coupled oscillators, which are now interpreted from the point of view of codimension-two bifurcations. On the Ott-Antonsen's manifold, complete bifurcation sets of delay-coupled Kuramoto model are derived regarding coupling strength and delay as bifurcation parameters. It is rigorously proved that the system must undergo Bautin bifurcations for some critical values, thus there always exists saddlenode bifurcation of periodic solutions inducing hysteresis loop. With the aid of center manifold reduction method and the Matlab Package DDE-Biftool, the location of Bautin and double Hopf points and detailed dynamics are theoretically determined. We find that, near these critical points, at most four coherent states (two of which are stable) and a stable incoherent state may coexist, and that the system undergoes Neimark-Sacker bifurcation of periodic solutions. Finally, the clear scenarios about the synchronous transition in delayed Kuramoto model are depicted.

Multiple Self-Locking in the Kuramoto--Sakaguchi System with Delay

“…Recently, also the version with time delay has been studied in some detail. In particular, the paper [13] pointed out the importance of codimension-two bifurcations, and [10] applied center manifold techniques to understand the bifurcation at the onset of synchronization.…”

Dynamics of disordered heterogeneous chains of phase oscillators

“…Multistability [Tilles et al, 2011;Girnyk et al, 2012;Niu, 2017], multiple basins of attractions [Delabays et al, 2017;Ha and Kang, 2012], and traveling waves [Hong and Strogatz, 2011] are some of the fundamental phenomena directly related to equilibrium propeerties of lattices of phase oscillators with both attractive and repulsive phase couplings. Among other papers we would like to mention very recent investigations [Matheny et al, 2019], where quasi-sinusoidal oscillators with linear nearest-neighbour coupling have been shown to manifest exotic regimes, such as splay states, inhomogeneous synchronization, clusters, and weak chimeras.…”