American Institute of Mathematical Sciences

November  2019, 24(11): 6025-6052. doi: 10.3934/dcdsb.2019119

Applying battelli-fečkan's method to transversal heteroclinic bifurcation in piecewise smooth systems

 1 School of Information Engineering, Southwestern University of Finance and Economics, Chengdu, Sichuan 610074, China 2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Zhengdong Du

Received  December 2018 Published  June 2019

Fund Project: The first author is supported by Humanities and Social Sciences Foundation of Ministry of Education of China under Grant Number 15YJAZH037. The second author is supported by NSFC (China) under Grant Number 11371264

In the last few years, Battelli and Fečkan 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.

Citation: Yurong Li, Zhengdong Du. Applying battelli-fečkan's method to transversal heteroclinic bifurcation in piecewise smooth systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6025-6052. doi: 10.3934/dcdsb.2019119
References:

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References:
A heteroclinic cycle $\Gamma$ of the unperturbed system (2) with $n = 2$ and $m = 4$
The heteroclinic cycle $\Gamma$ of the unperturbed system of (24) with $\lambda = 1.05$ and $\eta = 0.75$