一类双面碰撞振子的对称性尖点分岔与混沌  

Symmetry,Cusp Bifurcation and Chaos of an Impact Oscillator Between Two Rigid Sides

作　　者：乐源[1] 谢建华[1] 

机构地区：[1]西南交通大学应用力学与工程系,成都610031

出　　处：《应用数学和力学》2007年第8期991-998,共8页Applied Mathematics and Mechanics

基　　金：国家自然科学基金资助项目(10472096)

摘　　要：讨论了一类单自由度双面碰撞振子的对称型周期n-2运动以及非对称型周期n-2运动.把映射不动点的分岔理论运用到该模型,并通过分析对称系统的Poincaré映射的对称性,证明了对称型周期运动只能发生音叉分岔.数值模拟表明:对称系统的对称型周期n-2运动,首先由一条对称周期轨道通过音叉分岔形成具有相同稳定性的两条反对称的周期轨道;随着参数的持续变化,两条反对称的周期轨道经历两个同步的周期倍化序列各自生成一个反对称的混沌吸引子.如果对称系统演变为非对称系统,非对称型周期n-2运动的分岔过程可用一个两参数开折的尖点分岔描述,音叉分岔将会演变为一支没有分岔的分支以及另外一个鞍结分岔的分支.Both the symmetric period n-2 motion and asymmetric one of a one-degree-of-freedom impact oscillator were considered. The theory of bifurcations of the fixed point was applied to such model, and it was proved that the symmetric periodic motion only has pitchfork bifurcation by the analysis of the symmetry of the Poincaré map. The numerical simulation shows that one symmetnc periodic orbit could bifurcate into two antisymmetric ones via pitchfork bifurcation. While the control parameter changes continuously, the two antisymmetric periodic orbits will give birth to two synchronous antisymmetric period-doubling sequences, and bring about two antisymmetric chaotic atwactors subsequently. If the symmetric system is transformed into asymmetric one, bifurcations of the asymmetric period n-2 motion can be described by a two-parameter unfolding of cusp, and the pitchfork changes into one unbifurcated branch and one fold branch.

关 键 词：周期运动 POINCARÉ映射 对称性 音叉分岔 混沌 尖点 

分 类 号：O313.4[理学—一般力学与力学基础]