An approach to normal forms of Kuramoto model with distributed delays and the effect of minimal delay

Abstract

Heterogeneous delays with positive lower bound (gap) are taken into consideration in Kuramoto oscillators. We first establish a perturbation technique, by which universal normal forms and detailed dynamical behavior of this model can be obtained easily. Theoretically, a hysteresis loop is found near the subcritically bifurcated coherent state on the Ott-Antonsen's manifold. For Gamma distributed delay with fixed variance and mean, we find large gap destroys the loop and significantly increases in the number of coexisted coherent attractors. This result is also explained in the viewpoint of excess kurtosis.

Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear

“…In Section 4, the stability and direction of periodic solutions bifurcating from Hopf bifurcations are investigated by using the normal form theory and the center manifold theorem due to [31], which gives clearly the location where coherent states appear and their stability. In Section 5, inspired by the method given in [19,24,25], numerical simulations are carried out to support the obtained results.…”

“…show are about the bifurcation branches of the model, because bifurcation branches are widely used in literatures [19,24,25], which are also a quite unambiguous way to show the change of numbers of steady states or periodic oscillations.…”

“…Remark 2 Assume the incoherence is stable when τ = 0, and Hopf bifurcation occurs at τ . By using the global Hopf bifurcation results (see [19,22]), we know there will always exist hysteresis loop near the subcritical bifurcations. This will be shown in the simulation part such as Figure 3(a).…”

“…Thus 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 [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 General Non-Abelian Kuramoto Model on the 3-sphere

“…Another crucial assumption is that the coupling is global (or mean field coupling). This assumption allows time-delayed or phase-shifted coupling, but excludes coupling with distributed time delays [21,28,29] or coupling through complex networks [3,34].…”