一类碰撞振动系统的倍周期分岔研究

Study of Period-doubling Bifurcations of a Vibro-impact System

为了研究倍化分岔与Hopf分岔之间的联系,研究了一类碰撞振动系统因周期运动失稳而产生倍化分岔的问题。首先给出了该系统周期1-1运动的Poincaré映射建立过程,然后根据其映射的线性化矩阵的特征值穿越单位圆情况分析其映射不动点发生倍化分岔的可能性,最后通过数值计算加以验证。研究表明:系统存在典型倍周期分岔,另外单参数变化产生非共振条件下的Hopf分岔时,当参数进一步变化而越过共振点附近的某个共振区时,系统会产生非典型的倍周期分岔,其倍化分岔序列的分支数取决于强(弱)共振的阶数。

In order to study the relation of the period-doubhng bifurcation and the Hopf bifurcation, the period-doubling bifurcation problems of a vibro-impact system is investigated when the periodical solutions lose its stability. First, the process of establishing the Poincar6 map of the 1 - 1 motion of this system is provided. Second, the probability of the period-doubling bifurcation is analysed according to the Eigenvalue cross the unit circle. Finally, numerical simulation results demonstrate it. The typical period-doubling bifurcation is found. Mter the Hopf bifurcation in a non-resonance case, the non-typical period-doubling bifurcations occur while one single parameter varies crossing the resonance neighborhood. The number of orbits of the period-doubling bifurcation cascade depends on the order of strong （weak） resonance.