Delayed Hopf bifurcation in time-delayed slow-fast systems  

作　　者：ZHENG YuanGuang1 & WANG ZaiHua1,2 1 Institute of Vibration Engineering Research,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China 2 Institute of Science,PLA University of Science and Technology,Nanjing 211101,China 

出　　处：《Science China(Technological Sciences)》2010年第3期656-663,共8页中国科学（技术科学英文版）

基　　金：supported by the Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No.200430);in part by the National Natural Science Foundation of China (Grant Nos.10825207,10532050)

摘　　要：This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems.Here the two delayed's have different meanings.The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point,but at some other point which is above the bifurcation point by an obvious distance.In a time-delayed system,the evolution of the system depends not only on the present state but also on past states.In this paper,the time-delayed slow-fast system is firstly simplified to a slow-fast system without time delay by means of the center manifold reduction,and then the so-called entry-exit function is defined to characterize the delayed bifurcation on the basis of Neishtadt's theory.It shows that delayed Hopf bifurcation exists in time-delayed slow-fast systems,and the theoretical prediction on the exit-point is in good agreement with the numerical calculation,as illustrated in the two illustrative examples.This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems.Here the two delayed’s have different meanings.The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point,but at some other point which is above the bifurcation point by an obvious distance.In a time-delayed system,the evolution of the system depends not only on the present state but also on past states.In this paper,the time-delayed slow-fast system is firstly simplified to a slow-fast system without time delay by means of the center manifold reduction,and then the so-called entry-exit function is defined to characterize the delayed bifurcation on the basis of Neishtadt’s theory.It shows that delayed Hopf bifurcation exists in time-delayed slow-fast systems,and the theoretical prediction on the exit-point is in good agreement with the numerical calculation,as illustrated in the two illustrative examples.

关 键 词：time delay DELAYED BIFURCATION HOPF BIFURCATION slow-fast systems exit-point ENTRY-EXIT function 

分 类 号：O322[理学—一般力学与力学基础]