Consistency Study Of The Thermodynamic Kuramoto Model

The multi-agent model is a research hotspot in the field of control theory.The main issues studied in this field include consistency,containment control,and collective aggregation.The Kuramoto model is a special type of nonlinear multi-agent model.Due to its simplicity,it can explain many consistency problems related to models in various fields.However,a clear limitation of the Kuramoto model is that it does not use internal modes in its interaction(communication)mechanism.Since the Kuramoto model only involves the mechanical characteristics of particles.To better describe the dynamics of this active particle model,Ha et al.introduced a thermodynamic Kuramoto model(TK model)driven by a mixture theory of multiple-temperature gases.This paper,we study the TK model in more complex situations using stochastic differential equations,stochastic processes,and other methods.The main work is as follows:Firstly,the influence of some external fields on the cooperative behavior of the TK model is studied.The model is applied to two types of external fields:(a)white noise,(b)white noise,and periodic force.The cooperative behavior of the model under these two external fields is discussed.For models affected by additive white noise,it is proved that the coupling strength and noise strength between oscillators cannot promote the phase synchronization behavior of the model in the mean square sense.For models affected by multiplicative white noise,A sufficient condition for the phase synchronization behavior of the model is given.Similar conclusions were obtained for the TK model under the combination of periodic force and white noise: it is proved that the phase synchronization behavior of the model cannot occur in the mean square sense under the influence of additive noise and periodic force,while models affected by periodic forces and multiplicative white noise can achieve synchronization under certain conditions.Secondly,the cooperative behavior of the dynamically coupled TK model is studied.Considering the existence of dynamic coupling forces between oscillators,a dynamically coupled TK model is established.Whether the natural frequencies are the same is discussed separately.The sufficient conditions for the oscillators with the same natural frequency to exhibit asymptotically complete phase synchronization behavior are given.This article also provides a correlation theorem for making oscillators with different natural frequencies exhibit asymptotic phase locking.Furthermore,the similarities and differences between the dynamically coupled TK model,the TK model,and the modified Kuramoto model are compared,the theorem is extended to TK model and modified Kuramoto model.Finally,this paper extends the TK model to a more general situation,and establishes a dynamically coupled TK model affected by external fields.For a dynamically coupled TK model affected by additive white noise and a combined field containing additive white noise,phase synchronization behavior cannot occur in the mean square sense.In addition,A sufficient condition for phase synchronization of the model with multiplicative white noise is also given.