Synchronization Of Non-identical High-dimensional Kuramoto Model Based On Digraphs

Synchronization phenomena exist widely in nature,such as the convergent fish swarm,the glowing fireflies,the synchronization of neurons in the brain,the synchro-nization of multiple coupled pendulums and the synchronization of generators-Some meaningful research issues include how to establish a mathematical model to explain the synchronization phenomena in nature,how to design a synchronization mechianis-m and how to apply the synchronization mechanism to practice.Kuramoto model is one of the most famous mathematical models of synchronization and has been widely paid great attention in physics,biology and engineering.The traditional Kuramoto model is essentially a set of differential equations coupled by multiple individual dy-namic equations limited on the unit circle of the plane.High-dimensional Kuramoto model is a generalization of the traditional Kuramoto model limited on the unit sphere of a high-dimensional linear space.If the individuals of a high-dimensional Kuramo-to model have the identical dynamic equation,it is called identical Kuramoto model,otherwise it is called heterogeneous Kuramoto model.At present,the research re-sults of high-dimensional Kuramoto model are mostly concentrated on identical high-dimensional Kuramoto model,but few on heterogeneous the high-dimensional case.In this paper,the heterogeneous Kuramoto model in high-dimensional linear s-pace is studied.For the general interconnecting networks,the practical synchroniza-tion is proved under certain conditions.The dynamic system of synchronization error is deduced for the first time,which establishes a new theoretical framework for the study of synchronization problem.In particular,for the case of two individuals,the equilibrium points of synchronization error dynamic system are obtained and the cor-responding stability analysis is given.Finally,two numerical examples and some simulations are provided to verify the theoretical results.