碰撞振动系统四阶共振下的Hopf分岔和次谐分岔

建立了一类三自由度含间隙碰撞振动系统的力学模型,求解了系统六维n?1周期运动的周期解及其Poincaré映射。通过理论分析和数值模拟相结合的方法,分析了该系统在强共振点附近,系统两参数控制的局部动力学行为。即在两参数平面上共振点的附近变化两控制参数,进行数值模拟并划分两参数平面的拓扑区域;分析了以“四方形”和“四叶形”异宿轨道为特征的存在于强共振点附近的Hopf分岔不变圈和次谐分岔4?4周期运动,并进一步分析了四阶次谐分岔向混沌的演化过程。

HOPF AND SUBHARMONIC BIFURCATION OF VIBRO-IMPACT SYSTEM IN ORDER 4 RESONANCE CASE

A three-degree-of-freedom vibro-impact system is considered in this paper. Based on the solutions of differential equations between impacts, impact conditions and match conditions of periodic motion, the six-dimensional Poincaré maps of n ?1periodic motion are established. The two-parameter unfoldings of local dynamical behavior in resonance are investigated. By numerical simulation, as two controlling parameters varying on the two-parameter plane, the topology regions of parameter plane are divided. An invariant torus via Hopf bifurcation and period 4 ? 4 motions via subharmonic bifurcation are analyzed, which are characterized by “four-square” and “four-leaf ” different lodge orbits and exist near the critical point, and the route from order 4 subharmonic bifurcation to chaos is further analyzed.