Invariant polygons in systems with grazing-sliding.
Invariant polygons in systems with grazing-sliding.
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DOI：
10.1063/1.2904774
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发表时间：
2008-06
期刊：
影响因子：
2.9
通讯作者：
R. Szalai;H. Osinga
中科院分区：
文献类型：
--
作者：
R. Szalai;H. Osinga
本文研究了具有周期轨道的一般三维非光滑系统。我们假设周期轨道是不稳定的，具有复乘法器，因此系统中存在两个主频率。由于掠滑会导致维数损失，且不稳定性会导致每条轨迹都发生滑动，因此系统有一个由前向滑动轨道组成的吸引子。我们在一个适当选择的庞加莱截面上用一个可以看作是正规形式的三参数广义映射来分析这个吸引子。我们证明，在这个范式中，吸引子必须包含在多边形顶点相交的有限条直线中。然而，吸引子通常比相关的多边形大。我们将形成吸引子的线数作为参数的函数进行分类。进一步，在参数值固定的情况下，研究了吸引子的一维动力学。
The paper investigates generic three-dimensional nonsmooth systems with a periodic orbit near grazing-sliding. We assume that the periodic orbit is unstable with complex multipliers so that two dominant frequencies are present in the system. Because grazing-sliding induces a dimension loss and the instability drives every trajectory into sliding, the system has an attractor that consists of forward sliding orbits. We analyze this attractor in a suitably chosen Poincare section using a three-parameter generalized map that can be viewed as a normal form. We show that in this normal form the attractor must be contained in a finite number of lines that intersect in the vertices of a polygon. However the attractor is typically larger than the associated polygon. We classify the number of lines involved in forming the attractor as a function of the parameters. Furthermore, for fixed values of parameters we investigate the one-dimensional dynamics on the attractor.