一类碰撞振动系统的激变和拟周期-拟周期阵发性

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振动与冲击 ›› 2017, Vol. 36 ›› Issue (7) : 1-7.

论文

乐源1 缪鹏程2

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Crisis and quasiperiod-quasiperiod intermittency in a vibro-impact system

YUE Yuan,MIAO Pengcheng

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摘要

论文研究了一类三自由度碰撞振动系统的激变和阵发性。六维庞加莱映射能够表示成另外一个不对称映射的二次迭代，这表明系统具有对称性。该系统普遍存在发生Hopf分岔后得到的一对共轭拟周期运动。根据动力系统的极限集理论，讨论了极限集的对称性，得到系统发生激变的条件，并引入一个距离函数判定对称性恢复和激变临界点。当共轭混沌吸引子和不稳定对称不动点的最小距离等于零时，一对共轭混沌吸引子将会与不稳定的对称不动点在其吸引域边界发生碰撞，从而导致激变。通过数值模拟，揭示了激变之后的一种新的阵发性动力学现象：拟周期-拟周期阵发性。其分岔机制是：两个共轭拟周期吸引子→两个共轭拟周期吸引子倍化→两个共轭带状混沌吸引子→一个对称混沌吸引子→一个对称拟周期引子。通过对称极限集理论来区分对称吸引子和共轭吸引子，同时采用QR方法计算Lyapunov指数并用来确定吸引子的类型。激变导致的拟周期-拟周期阵发性，对于多自由度碰撞振动系统的动力学研究及优化设计具有重要意义。

Abstract

Crisis and quasiperiod-quasiperiod intermittency in a 3-DOF vibro-impact system with symmetry were studied.The system’s 6-dimensional Poincaré map was expressed as the second iteration of another unsymmetric map,it implied that the system has a symmetry.Two conjugate quasi-periodic motions,coming from two conjugate periodic motions after Hopf bifurcation coexisted widely in such a dynamic system.According to the limit set theory of dynamic systems and the symmetry of the limit set,a distance function was introduced to detect the crisis of symmetry increasing.It was shown that when the minimum distance between a pair of conjugate chaotic attractors and an unstable symmetric fixed point is close to zero,a pair of conjugate chaotic attractors do not collide with the unstable symmetric fixed point on the attracting field boundary,to lead to a crisis.Numerical simulations revealed that a new intermittency behavior named the quasiperiod-quasiperiod intermittency occurs; the mechanism of symmetry restoring of quasi-periodic motion is two conjugate tori (quasi-periodic) → doubling of two conjugate tori → two conjugate band chaos attractors → a pair of symmetric chaos attractors → one symmetric torus (quasi-periodic); the symmetric limit set is introduced to distinguish symmetric attractors from conjugate ones; Lyapunov exponent spectrum computed with QR method is used to determine the type of attractors; the quasiperiod-quasiperiod intermittency is of importance for the optimization design of vibro-impact systems.

关键词

碰撞振动系统 / 拟周期运动 / 激变 / 阵发性

Key words

vibro-impact system

/ quasi-periodic motion / crisis / intermittency

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乐源1 缪鹏程2. 一类碰撞振动系统的激变和拟周期-拟周期阵发性[J]. 振动与冲击, 2017, 36(7): 1-7

YUE Yuan,MIAO Pengcheng. Crisis and quasiperiod-quasiperiod intermittency in a vibro-impact system[J]. Journal of Vibration and Shock, 2017, 36(7): 1-7

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