Invariant cone and synchronization state stability of the mean field models

Abstract

In this article we prove the stability of some mean field systems similar to the Winfree model in the synchronized state. The model is governed by the coupling strength parameter κ and the natural frequency of each oscillator. The stability is proved independently of the number of oscillators and the distribution of the natural frequencies.The main result is proved using the positive invariant cone method for the linearized system. This method can be applied to other mean field models as in the Kuramoto model.

Exponential stable manifold for some linear system with application to the mean field models

“…In 1967 Winfree [9] proposed a mean eld model describing the synchronization of a population of organisms or oscillators that interact simultaneously [1,2,3,4,5,7].…”

“…The main idea of the proof is to rewrite the linearized Winfree model by making a change of time t ↔ s and a linear change of the tangent vectors y ↔ z. In Lemme 3.1 of [7] we proved that the velocity of µ is strictly positive,…”

Collective behaviors of a Winfree ensemble on an infinite cylinder

“…In [15,16], the robustness of the COD state has been investigated in general network under the time-delayed interactions, and the interplay of adaptive coupling and random effect, respectively. The existence of periodic locked orbit and complete phase-locking were studied in [28,29].…”

“…The uniform stability in Theorem 3.5 implies that the sequence {µ N ∞ } is a Cauchy sequence and thus generates a unique limit measure µ ∞ . Moreover, estimate (29) gives…”