Hopf-flip bifurcation of high dimensional maps and application to vibro-impact systems  

作　　者：Jianhua Xie Wangcai Ding E.H. Dowell L. N. Virgin 

机构地区：[1]Department of Applied Mechanics and Engineering,Southwest Jiaotong University, Chengdu 610031, China [2]School of Mechanical Engineering, Lanzhou Jiaotong University,Lanzhou 730070, China [3]School of Engineering.Duke University. Durham, NC27708-0300,USA

出　　处：《Acta Mechanica Sinica》2005年第4期402-410,共9页力学学报（英文版）

基　　金：The project supported by the Nutional Natural Science Foundation of China(10472096)

摘　　要：This paper addresses the problem of Hopf-flip bifurcation of high dimensional maps. Using the center manifold theorem, we obtain a three dimensional reduced map through the projection technique. The reduced map is further transformed into its normal form whose coefficients are determined by that of the original system. The dynamics of the map near the Hopf-flip bifurcation point is approximated by a so called “time-2τ^2 map” of a planar autonomous differential equation. It is shown that high dimensional maps may result in cycles of period two, tori T^1 （Hopf invariant circles）, tori 2T^1 and tori 2T^2 depending both on how the critical eigenvalues pass the unit circle and on the signs of resonant terms＇ coefficients. A two-degree-of-freedom vibro-impact system is given as an example to show how the procedure of this paper works. It reveals that through Hopf-flip bifurcations, periodic motions may lead directly to different types of motion, such as subharmonic motions, quasi-periodic motions, motions on high dimensional tori and even to chaotic motions depending both on change in direction of the parameter vector and on the nonlinear terms of the first three orders.This paper addresses the problem of Hopf-flip bifurcation of high dimensional maps. Using the center manifold theorem, we obtain a three dimensional reduced map through the projection technique. The reduced map is further transformed into its normal form whose coefficients are determined by that of the original system. The dynamics of the map near the Hopf-flip bifurcation point is approximated by a so called “time-2τ^2 map” of a planar autonomous differential equation. It is shown that high dimensional maps may result in cycles of period two, tori T^1 （Hopf invariant circles）, tori 2T^1 and tori 2T^2 depending both on how the critical eigenvalues pass the unit circle and on the signs of resonant terms＇ coefficients. A two-degree-of-freedom vibro-impact system is given as an example to show how the procedure of this paper works. It reveals that through Hopf-flip bifurcations, periodic motions may lead directly to different types of motion, such as subharmonic motions, quasi-periodic motions, motions on high dimensional tori and even to chaotic motions depending both on change in direction of the parameter vector and on the nonlinear terms of the first three orders.

关 键 词：MAPS Vibro-impact dynamics Hopf-flip bifurcation TORUS CHAOS 

分 类 号：O32[理学—一般力学与力学基础]