Research On Synchronous Dynamics Of Cartesian Product Of Kuramoto Models

In a population of interacting elements,synchronous behavior occurs spontaneously,which is an important and worthy research topic in nonlinear dynamics and complex networks.Simply speaking,synchronization refers to the formation of consistent rhythms between two or more individuals because the connections between individuals adjust themselves to from a consistent state,while central regulations are often absent.Studying the internal microscopic mechanism of such phenomena provides insights into collective behaviors in a wide range of fields,such as the simultaneous flashing of male fireflies,circadian rhythms,and widespread applause.In the theoretical study of synchronization,the most successful model proposed is the Kuramoto model,which represents a classic example of synchronization.Although this model of coupled phase oscillators is simple in form,it captures the essence of the synchronization process which can be solved analytically.In the past few decades,the Kuramoto model and generations have inspired and simulated extensive research in many areas,including basic theoretical analysis and related practical applications.Later,with the rapid development of complex networks,it has become a hot issue to study the synchronization behavior of coupled phase oscillators on the basis of complex networks.It is found that the topologies of networks often affect the synchronization of coupled phase oscillator subsystems.The star network is one of the basic motifs for constructing complex network topology and dynamics,which has been largely reveal the dynamic mechanism of explosive synchronization.Starting from the synchronous dynamics composed of star networks.This paper propose to construct complex dynamic phase transition processes as observed in nature.More specifically,we propose to construct complex synchronization dynamics by the Cartesian product of Kuramoto models on two independent star networks.In this product model,we use Watanabe-Strogatz method to completely solve the low-dimensional dynamic equation of the global order parameters which describe collective dynamics.Due to the different choices of network parameters,the hysteresis areas of two independent factor networks have three different types of interaction scenarios,which further changes the attraction basin of multiple fixed points.More importantly,when the oscillators on one star network are synchronized and the oscillators on the other star network are not synchronized,we can get the cluster synchronization state on the product network,and the number of clusters is completely controlled.Our numerical simulation results are in complete agreement with theoretical predictions.Our work provides a novel "bottom-up" framework to generate more complex dynamic behaviors encountered in actual complex systems.