Applying battelli-fečkan's method to transversal heteroclinic bifurcation in piecewise smooth systems
Applying battelli-fečkan's method to transversal heteroclinic bifurcation in piecewise smooth systems
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DOI：
10.3934/dcdsb.2019119
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发表时间：
2017
影响因子：
1.2
通讯作者：
Yurong Li;Zhengdong Du
中科院分区：
文献类型：
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作者：
Yurong Li;Zhengdong Du
在过去的几年里，Battelli 和 Feckan 开发了一种泛函分析方法来严格证明时间扰动分段平滑系统中混沌行为的存在，该系统的未扰动部分具有分段连续同宿解。在本文中，通过应用他们的方法，我们研究了时间扰动分段光滑系统中的混沌现象，该系统在有限多个切换流形上具有不连续性，其未扰动部分在每个子区域中都有一个双曲鞍点，以及连接这些鞍点的异宿轨道，该异宿轨道横向穿过每个切换流形一次。我们得到一组梅尔尼科夫型函数，其零点对应于系统混沌的发生。此外，还明确给出了平面分段光滑系统的梅尔尼科夫函数。作为一个应用，我们提出了一个具有四个区域的准周期激励三维分段线性系统的示例。
In the last few years, Battelli and Feckan have developed a functional analytic method to rigorously prove the existence of chaotic behaviors in time-perturbed piecewise smooth systems whose unperturbed part has a piecewise continuous homoclinic solution. In this paper, by applying their method, we study the appearance of chaos in time-perturbed piecewise smooth systems with discontinuities on finitely many switching manifolds whose unperturbed part has a hyperbolic saddle in each subregion and a heteroclinic orbit connecting those saddles that crosses every switching manifold transversally exactly once. We obtain a set of Melnikov type functions whose zeros correspond to the occurrence of chaos of the system. Furthermore, the Melnikov functions for planar piecewise smooth systems are explicitly given. As an application, we present an example of quasiperiodically excited three-dimensional piecewise linear system with four zones.