The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold
The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold
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用于检测具有切换流形的平面混合分段平滑系统中的混沌动力学的梅尔尼科夫方法
DOI：
10.1007/s11071-017-3493-2
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发表时间：
2017
影响因子：
5.6
通讯作者：
Yuxin Hao
中科院分区：
文献类型：
--
作者：
Shuangbao Li;Xiaojun Gong;Wei Zhang;Yuxin Hao
本文将光滑系统的经典Melnikov方法推广到一类受时间周期扰动的平面混合分段光滑系统。在这一类中，我们假设存在一个更一般形式的切换流形，使得平面被划分为两个区域，并且每个区域的动力学由一个光滑系统控制。进一步，我们假设无摄动系统是一般平面的非零迹分段光滑系统，并且具有横过开关流形的分段光滑同斜轨道。我们还定义了一个重置映射来描述轨迹到达交换流形时对交换流形的瞬时冲击规律。通过一系列的几何分析和摄动技术，我们得到了一个melnikov型函数来测量在时间周期摄动和重置映射的作用下不稳定流形和稳定流形的分离。最后，我们利用所提出的Melnikov函数研究了具体平面分段线性振荡器的全局分岔和混沌动力学。
In this paper, we extend the classical Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth system subjected to a time-periodic perturbation. In this class, we suppose there exists a switching manifold with a more general form such that the plane is divided into two zones, and the dynamics in each zone is governed by a smooth system. Furthermore, we assume that the unperturbed system is a general planar piecewise-smooth system with non-zero trace and possesses a piecewise-smooth homoclinic orbit transversally crossing the switching manifold. We also define a reset map to describe the instantaneous impact rule on the switching manifold when a trajectory arrives at the switching manifold. Through a series of geometrical analysis and perturbation techniques, we obtain a Melnikov-type function to measure the separation of the unstable manifold and stable manifold under the effect of the time-periodic perturbations and the reset map. Finally, we use the presented Melnikov function to study global bifurcations and chaotic dynamics for a concrete planar piecewise-linear oscillator.