Low-dimensional maps for piecewise smooth oscillators
Low-dimensional maps for piecewise smooth oscillators
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DOI：
10.1016/j.jsv.2007.04.044
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发表时间：
2007-09
影响因子：
4.7
通讯作者：
E. Pavlovskaia;M. Wiercigroch
中科院分区：
文献类型：
--
作者：
E. Pavlovskaia;M. Wiercigroch
研究了分段光滑非线性振子的动力学问题，提出了将多维流降维为低维映射的一般方法。这包括分段光滑振子的定义和创建提供精确解的全局迭代映射。全局映射由局部映射组成，局部映射在相空间的光滑子区域中构造。为了构造这种低维映射，建议通过轨迹来监视光滑子空间之间的选定边界的交点。降维效果与所选边界的维数直接相关，边界维数越低，降维效果越好。详细讨论了漂移碰撞振子，将五维流动简化为一维近似解析映射。首先，一个确切的二维地图已制定和分析。进一步减少到一维近似映射的介绍和讨论。标准的非线性动力学分析揭示了一个复杂的行为，从周期振荡到混沌，并共存的多个吸引子。构造的地图的准确性进行检查，通过比较与精确的解决方案，在很宽的范围内的系统参数。
Dynamics of the piecewise smooth nonlinear oscillators is considered, for which, general methodology of reducing multidimensional flows to low-dimensional maps is proposed. This includes a definition of piecewise smooth oscillator and creation of a global iterative map providing an exact solution. The global map is comprised of local maps, which are constructed in the smooth sub-regions of phase space. To construct this low-dimensional map, it is proposed to monitor the points of intersections of a chosen boundary between smooth subspaces by a trajectory. The dimension reduction is directly related to the dimension of the chosen boundary, and the lower its dimension is, the larger dimension reduction can be achieved. Full details are given for a drifting impact oscillator, where the five-dimensional flow is reduced to one-dimensional (1D) approximate analytical map. First an exact two-dimensional map has been formulated and analysed. A further reduction to 1D approximate map is introduced and discussed. Standard nonlinear dynamic analysis reveals a complex behaviour ranging from periodic oscillations to chaos, and co-existence of multiple attractors. Accuracy of the constructed maps is examined by comparing with the exact solutions for a wide range of the system parameters.