多自由度含间隙振动系统周期运动的二重Hopf分岔

DOUBLE HOPF BIFURCATION OF PERIODIC MOTION OF THE MULTI-DEGREE-OF-FREEDOM VIBRATORY SYSTEM WITH A CLEARANCE

基于Poincaré映射方法和数值仿真分析了多自由度含间隙振动系统对称型周期碰撞运动的稳定性与二重Hopf分岔。应用映射的中心流形和范式方法,研究了高维映射在其Jacobian矩阵两对复共轭特征值同时穿越复平面单位圆周情况下的余维二分岔,分析了映射在二重Hopf分岔点附近的双参数开折,揭示了含间隙振动系统在二重Hopf分岔点附近的动力学行为。含间隙振动系统在二重Hopf分岔点附近存在对称型周期碰撞运动、对称型周期碰撞运动的Hopf分岔、环面分岔及“轮胎”型概周期吸引子。环面分岔导致了半吸引不变环和复杂的“轮胎”型概周期吸引子。

A multi-degree-of-freedom vibratory system having symmetrically placed rigid stops and subjected to periodic excitation is considered. Local codimension two bifurcation of the vibro-impact system, concerning two complex conjugate pairs of eigenvalues of linearized map escaping the unit circle simultaneously, is analyzed using the center manifold theorem and normal form method of maps. Local behavior of the system, near the point of double Hopf bifurcation, is investigated using qualitative analysis and numerical simulation. Near the value of double Hopf bifurcation there exist period-one double-impact symmetrical motion, Hopf bifurcation and toms bifurcation. The quasi-periodic impact motions are represented by the closed circle and ＂tire-like＂ a^actor in projected Poincaré sections. With change of system parameters, the quasi-periodic impact motions usually lead to chaos via ＂tire-like＂ tori doubling.