Abstract
In this paper, time delay effect and distributed shear are considered in the Kuramoto model. On the Ott-Antonsen’s manifold, through analyzing the associated characteristic equation of the reduced functional differential equation, the stability boundary of the incoherent state is derived in multiple-parameter space. Moreover, very rich dynamical behavior such as stability switches inducing synchronization switches can occur in this equation. With the loss of stability, Hopf bifurcating coherent states arise, and the criticality of Hopf bifurcations is determined by applying the normal form theory and the center manifold theorem. On one hand, theoretical analysis indicates that the width of shear distribution and time delay can both eliminate the synchronization then lead the Kuramoto model to incoherence. On the other, time delay can induce several coexisting coherent states. Finally, some numerical simulations are given to support the obtained results where several bifurcation diagrams are drawn, and the effect of time delay and shear is discussed.
Highlights
The Kuramoto model was first proposed in Refs. 1 and 2, consisting of a group of phase oscillators
Through some rigorous bifurcation analysis, we find that time delay can have very important impact on the system dynamics: compared with the previous results in Refs. 16–18, time delay will further prevent synchronization in some cases and will induce several coexisting coherent states
As shown in figure (b), we find that the Kuramoto model exhibits stable coherent states for any τ when γ is small
Summary
The Kuramoto model was first proposed in Refs. 1 and 2, consisting of a group of phase oscillators. As stated in Ref. 16, the synchronization transition in (1) fails in case of large spread of shear distribution in the absence of time delay. As known to all, time delay will induce synchronization transition in Kuramoto model.[27,28] in this paper we will study the effect of delay together with shear on the synchronization transition from the bifurcation approach. The idea is followed by Ref. 19 based on the Ott-Antonsen’s manifold reduction.[32,33] Kuramoto model (1) can be reduced into a functional differential equation on a submanifold of the phase space. The transition can be described by Hopf bifurcations which will be discussed in the parameter space consisting of delay and shear.
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