Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics

Abstract

We study the practical synchronization of the Kuramoto dynamics of units distributed over networks. The unit dynamics on the nodes of the network are governed by the interplay between their own intrinsic dynamics and Kuramoto coupling dynamics. We present two sufficient conditions for practical synchronization under homogeneous and heterogeneous forcing. For practical synchronization estimates, we employ the configuration diameter as a Lyapunov functional, and derive a Gronwall-type differential inequality for this value.2010 Mathematics Subject Classification. Primary: 70F99; Secondary: 92B25.

The Wigner-Lohe model for quantum synchronization and its emergent dynamics

“…Then, we take the inner product of the above relation with ψ i and compare the real part of the resulting relation to get the Kuramoto model for classical synchronization [1,6,13,16,17,18]:θ…”

Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring

“…The distance between i and j is the number of arcs in a shortest path connecting i and j. We now state a lemma from [10] as follows.…”

From approximate synchronization to identical synchronization in coupled systems

“…The studies on practical synchronization of coupled nonidentical systems with linearly diffusive couplings can be found in [36,38]. Frequency synchronization and phase synchronization of phase models, such as the Kuramoto model, have been investigated in [16,22,53].…”

“…, then applying the mean-value theorem yields the existence of some ψ j,k between φ j,k (θ) and φ j,k (ϑ), and some u j,k between θ and ϑ, with (5), (16), and (42). This justifies (44).…”

“…= (σ + ρ)(β + ϑ) 1/(2βσ),r 2 := (σ + ρ)(β + ϑ) 1/(2β),r 3 := (1 + √ 2)(β + ϑ)(σ + ρ)/(2β),ř 3 := [β − √ 2(β + ϑ)](σ + ρ)/(2β).The proof of this proposition is arranged in Appendix D. In what follows, we consider system (99) which satisfies assumption (D), with N = 2, K = 3, and Q =Q (ς) , for an arbitrarily fixed ς > 0. With (104),Ḡ k ,Ď k , andD k defined in (14)-(16) are nowḠ k =R (ς) k ,Ď k =D k = 1, k = 1, 2, 3. (105) For any ς > 0, system (99) satisfies assumption (E) withδ 1 = δ 2 = 0, andδ 3 = ϑ ·R in Proposition 4.1.…”