一类三自由度碰撞振动系统的Poincaré映射的对称性,分岔及混沌

Symmetry Bifurcation and Chaos of the Poincaré Map,in a Three-degree-of-freedom Vibro-impact System

考虑了一类具有对称刚性约束的三自由度碰撞振动系统。建立了系统的Poincaré映射,并导出了Poincaré映射的对称性。把映射不动点的稳定性与分岔理论应用于该系统,分析表明Poincaré映射的对称性完全抑制了对称周期n-2运动的周期倍化分岔,Hopf-flip分岔和pitchfork-flip分岔,并证明了两个反对称的周期n-2运动具有相同的稳定性。数值模拟得到了对称周期n-2运动的音叉分岔,Hopf分岔和Hopf-Hopf分岔。此外,通过Poincaré截面投影相图的形式研究了由音叉分岔通向混沌的路径。

A three-degree-of-freedom vibro-impact system with symmetric constraining stops is considered. The Poincaré map of the system is established, and the symmetry of the Poincaré map is derived in detail. The theory of bifurcation of fixed points is applied to such model, and it is shown that the symmetry of the Poincaré map suppres- ses codimension-1 period-doubling bifurcation, Hopf-flip bifurcation and pitchfork-flip bifurcation of symmetric period n - 2 motions. It is also proved that both the two antisymmetric period n - 2 motions have the same stability. By numerical simulations, pitchfork bifurcation, Hopf bifurcation and Hopf-Hopf bifurcation of symmetric period n - 2 motions are represented. Besides, the routes to chaos after pitchfork bifurcation are studied in the forms of the phase portrait in the projected Poincaré section.