Szalai R, Osinga H M
Bristol Centre for Applied Nonlinear Mathematics, University of Bristol, University Walk, Bristol, BS8 1TR, United Kingdom.
Chaos. 2008 Jun;18(2):023121. doi: 10.1063/1.2904774.
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.
本文研究了在擦边滑动附近具有周期轨道的一般三维非光滑系统。我们假设该周期轨道具有复乘数从而不稳定，使得系统中存在两个主导频率。由于擦边滑动会导致维度损失且不稳定性会使每个轨迹进入滑动状态，该系统具有一个由向前滑动轨道组成的吸引子。我们使用一个可视为范式的三参数广义映射，在适当选择的庞加莱截面中分析这个吸引子。我们表明，在这种范式中，吸引子必定包含在有限数量的直线中，这些直线在一个多边形的顶点处相交。然而，吸引子通常比相关多边形更大。我们将形成吸引子的直线数量作为参数的函数进行分类。此外，对于固定的参数值，我们研究吸引子上的一维动力学。
