# American Institute of Mathematical Sciences

October  2021, 26(10): 5581-5599. doi: 10.3934/dcdsb.2020368

## Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve

 1. School of Mathematics and Statistics, Anhui Normal University, Wuhu, Anhui, 241000, China a. Department of Mathematics Zhejiang Normal University Jinhua, Zhejiang, 321004, China b. Department of Mathematics Shanghai Normal University Shanghai, 200234, China

* Corresponding author: M. Han

Received  June 2019 Revised  July 2020 Published  December 2020

Fund Project: H. Tian is supported by National Natural Science Foundation of China (No.12001012), Natural Science Foundation of Anhui Province (No. 2008085QA10) and Scientific Research Foundation for Scholars of Anhui Normal University. M. Han is supported by National Natural Science Foundation of China (Nos. 11931016 and 11771296)

This paper deals with the number of limit cycles for planar piecewise smooth near-Hamiltonian or near-integrable systems with a switching curve. The main task is to establish a so-called first order Melnikov function which plays a crucial role in the study of the number of limit cycles bifurcated from a periodic annulus. We use the function to study Hopf bifurcation when the periodic annulus has an elementary center as its boundary. As applications, using the first order Melnikov function, we consider the number of limit cycles bifurcated from the periodic annulus of a linear center under piecewise linear polynomial perturbations with three kinds of quadratic switching curves. And we obtain three limit cycles for each case.

Citation: Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5581-5599. doi: 10.3934/dcdsb.2020368
##### References:

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##### References:
The orbit $\widehat{AA_\epsilon}$ of system (4)
The orbit $\widehat{AA_\epsilon}$ of system (31)
Periodic orbits and switching curve of system (34)$|_{\epsilon = 0}$
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