Dynamics Of A Class Of Singularly Perturbed Flocking Models

Singular perturbation models are a wide range of problems in the physical sciences such as physics,biology,and applied mathematics.For example,the sun-earth-lunar three-body problem in astronomical mechanics,the boundary layer problem in fluid mechanics,and virus propagation under extreme conditions,and so on can be regarded as different forms of singular perturbation problems.As early as the end of the 19th century,Poincareused the "small parameter method" and power series theory to systematically study the three-body problem,which opened the research history of the singular perturbation theory.The main idea of this theory is that for a class of mathematical problems with small parameters,the system’s time scale characteristics are used to decompose the system into two or more low-order systems,and then find the corresponding low-order system solution,and use them to construct and explain the changing phenomenon of the dominant part of the original system.To date,the research and application of singular perturbation theory has covered almost all subject areas,including boundary layers,reaction diffusion,quantum mechanics,plasma physics,thermodynamics,marine science,ecology,engineering,and social economics aspect and so on.In recent years,as people continue to study the phenomenon of flocking,flocking models of many different scales have been proposed,and they have attracted great attention from biologists and applied mathematicians,an important work in this regard is that in 2012,Seung-Yeal Ha et al.Studied the perturbation problem of adding small parameters to the classic Cucker-Smale model,the author uses the classic singular perturbation theory to obtain the flocking conclusion of the system under certain conditions,that is,the speed of the interacting individuals will eventually reach a consensus.The work of this paper is based on the work of Seung-Yeal Ha et al.We will discuss a flocking problem for a class of Cucker-Smale model with Rayleigh friction term and small parameters.We will combine the classical dynamic system theory,that is,the singular perturbation measure theory method,to conduct a more in-depth analysis of the local dynamics of the clustering state of the system under consideration.The full text is divided into three chapters.The first chapter is about introduction,which briefly introduces the background and main results of the research questions in this article;The second chapter is preliminaries,which briefly introduces the basic concepts and theories needed in this article;The third chapter is the main part of the article,by useing singular perturbation theory to discuss the dynamic properties of the Cucker-Smale model with Rayleigh friction,the flocking form solution is obtained and its local stability is proved.