Exponential synchronization of the high-dimensional Kuramoto model with identical oscillators under digraphs

Abstract

For the Kuramoto model and its variations, it is difficult to analyze the exponential synchronization under the general digraphs due to the lack of symmetry. In this paper, for the high-dimensional Kuramoto model of identical oscillators, a matrix Riccati differential equation (MRDE) is proposed to describe the error dynamics. Based on the MRDE, the exponential synchronization is proved by constructing a total error function for the case of digraphs admitting spanning trees. Finally, some numerical simulations are given to illustrate the obtained theoretical results.

Synchronization of high‐dimensional Kuramoto models with nonidentical oscillators and interconnection digraphs

“…In the following, we use ( 6) to investigate the practical synchronization of the high-dimensional Kuramoto model. [29]) Consider a sequence of unit vectors r 1 , r 2 , … , r m ∈ R n . Let e i j be defined by ( 4) for any i, j = 1, 2, … m. Then…”

“…Moreover, we have a positively invariant compact set S 𝜀 p described by (29). Thus, by the LaSalle invariance principle, r (t ) converges to a synchronization point (𝜉 T , 𝜉 T , … , 𝜉 T ) T , which implies the complete synchronization is achieved.…”

“…Lemma 4. (Lemma 3.2 of [29]) Assume that the digraph  has a directed spanning tree with weighted adjacency matrix A = (a i j ) ∈ R m×m . Then (i) there exists a constant c 1 > 0 such that…”

“…where r i ∈ R n is the state of the i th oscillator, Ω i is an n × n skew-symmetric matrix, k > 0 is the interconnection gain, and A = (a i j ) ∈ R m×m is the weighted adjacency matrix of the interconnecting graph. There are many theoretical results on synchronization behaviors of the high-dimensional Kuramoto model and its variations with identical oscillators [23][24][25][26][27][28][29]. In [30], Ha et [34].…”

Higher-order synchronization on the sphere

“…The system (2) has been intensively investigated for general d, usually with global coupling and for the homogeneous case Ω i = 0, whether as a model of synchronization [24][25][26][27][28][29][30], or opinion formation and consensus studies on the unit sphere [31][32][33][34][35][36][37], or for the modelling of swarming behaviour [16,38,39]. Many synchronization properties have been established for any d [25,26,[40][41][42], in particular for the case of identical frequencies Ω i = Ω, it is known that for κ 2 > 0 the order parameter r = X av evolves exponentially quickly to the value r ∞ = 1, in which case all nodes are co-located to form a completely synchronized state, or for κ 2 < 0 to a state with r ∞ = 0.…”