碰撞振动系统分岔与混沌的研究进展

针对工程实际中普遍存在的碰撞振动系统这种典型的非光滑动力系统, 其研究具有 重要的理论意义和工程实用价值. 碰撞振动系统动力学的分析与研究方法主要有理 论分析、数值模拟以及应用与实验研究. 为了研究碰撞振动系统的周期运动稳定性、 分岔及混沌, 采用的手段有建立Poincar\'{e}映射、中心流形和范式方法, 映射的 分岔与混沌理论是碰撞振动系统研究的理论基础. 首先简述了碰撞振动系统的分析 与研究方法, 光滑非线性系统动力学的分析方法部分可以推广到碰撞振动系统, 碰 撞振动的不连续性导致一些方法的适用性和有效性问题. 进一步综述了碰撞振动系 统周期运动稳定性、分岔、混沌及奇异性的理论研究和工程应用现状. 最后着重结 合相关离散型映射系统的动力学发展, 对碰撞振动系统的分岔与混沌研究及存在的 主要问题进行了讨论, 并展望了其发展趋势.

ADVANCES OF RESEARCH ON BIFURCATIONS AND CHAOS IN VIBRO-IMPACT SYSTEM

The vibro-impact system found in many practical systems, as a typical non-smooth dynamical system is important in engineering. Its studies include theoretical analysis, numerical simulations, applied and experimental studies. By establishing the Poincar\'{e} map, using the center manifold theorem and the theory of normal forms, the periodic motion stability, bifurcation and chaos of the vibro-impact system can be investigated. The theory of bifurcation and chaos of map is one of the bases in studying vibro-impact systems. The smooth nonlinear dynamics can be partly extended to vibro-impact systems, but the discontinuity caused by impact will affect applicability and effectiveness of some methods. The theoretical studies and engineering applications of stability, bifurcations, chaos and singularity of vibro-impact systems are surveyed in this paper. Finally, the discussion centers around some unsolved problems of bifurcations and chaos of vibro-impact systems, and the advance in the corresponding discrete map dynamics. The future research trends are suggested.