Melnikov Method for a Class of Planar Hybrid Piecewise-Smooth Systems
Melnikov Method for a Class of Planar Hybrid Piecewise-Smooth Systems
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DOI：
10.1142/s0218127416500309
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发表时间：
2016-03
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影响因子：
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通讯作者：
Shuangbao Li;Wensai Ma;Wei Zhang;Y. Hao
中科院分区：
文献类型：
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作者：
Shuangbao Li;Wensai Ma;Wei Zhang;Y. Hao
本文将光滑系统的Melnikov方法推广到一类周期摄动平面混合分段光滑系统。在这一类中，切换流形是一条将平面分成两个区域的直线，每个区域中的动力学由一个光滑系统来管理。当轨迹到达分隔线时，在进入另一个区域的轨迹之前，立即应用重置贴图。我们假设未扰动系统是一个分段哈密顿系统，它具有横跨切换流形的分段光滑同宿解。然后，我们研究了在非自治周期扰动和重置映射下同宿轨道的持久性。为了达到这一目的，我们得到了度量扰动稳定流形和不稳定流形之间距离的Melnikov函数，并给出了一类平面混合分段光滑系统的同宿分支定理。此外，我们利用所得到的Melnikov函数来检测具体平面混合分段光滑系统的混沌边界。
In this paper, we extend the well-known Melnikov method for smooth systems to a class of periodic perturbed planar hybrid piecewise-smooth systems. In this class, the switching manifold is a straight line which divides the plane into two zones, and the dynamics in each zone is governed by a smooth system. When a trajectory reaches the separation line, then a reset map is applied instantaneously before entering the trajectory in the other zone. We assume that the unperturbed system is a piecewise Hamiltonian system which possesses a piecewise-smooth homoclinic solution transversally crossing the switching manifold. Then, we study the persistence of the homoclinic orbit under a nonautonomous periodic perturbation and the reset map. To achieve this objective, we obtain the Melnikov function to measure the distance of the perturbed stable and unstable manifolds and present the theorem for homoclinic bifurcations for the class of planar hybrid piecewise-smooth systems. Furthermore, we employ the obtained Melnikov function to detect the chaotic boundaries for a concrete planar hybrid piecewise-smooth system.