BIFURCATION ANALYSIS IN A GENERALIZED FRICTION MODEL WITH TIME-DELAYED FEEDBACK  

作　　者：Chuang Xu Junjie Wei 

机构地区：[1]Dept. of Math., Harbin Institute of Technology, Harbin 150001 [2]Dept. of Mathematical and Statistical Sciences, University of Alberta,Edmonton, Alberta, Canada T6G 2G1

出　　处：《Annals of Differential Equations》2014年第2期199-215,共17页微分方程年刊（英文版）

基　　金：supported by the National Natural Science Foundation of China(No.11031002);Research Fund for the Doctoral Program of Higher Education of China(No.20122302110044)

摘　　要：In this paper, we consider a generalized model of the two friction models, both of which have two different types of control forces with time-delayed feedback proposed by Ashesh Sara et al. By taking the time delay as the bifurcation parameter, we discuss the local stability of the Hopf bifurcations. Under some condition, the generalized model harbors a phenomenon that the equilibrium may undergo finite switches from stability to instability to stability and finally become unstable. By applying the method introduced by Faria and Magalhaes, we compute the normal form on the center manifold to determine the direction and stability of the Hopf bifurcations. Numerical simulations are carried out and more than one periodic solutions may exist according to the bifurcation diagram given by BIFTOOL. Finally a brief conclusion is presented.In this paper, we consider a generalized model of the two friction models, both of which have two different types of control forces with time-delayed feedback proposed by Ashesh Sara et al. By taking the time delay as the bifurcation parameter, we discuss the local stability of the Hopf bifurcations. Under some condition, the generalized model harbors a phenomenon that the equilibrium may undergo finite switches from stability to instability to stability and finally become unstable. By applying the method introduced by Faria and Magalhaes, we compute the normal form on the center manifold to determine the direction and stability of the Hopf bifurcations. Numerical simulations are carried out and more than one periodic solutions may exist according to the bifurcation diagram given by BIFTOOL. Finally a brief conclusion is presented.

关 键 词：Hopf bifurcation friction-driven oscillation time-delayed feedback periodic solution complex dynamical behavior 

分 类 号：O175[理学—数学]