Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp

Jihua Yang; Erli Zhang; Mei Liu

doi:10.3934/cpaa.2017114

Article Contents

2017, Volume 16, Issue 6: 2321-2336. Doi: 10.3934/cpaa.2017114

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Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp

1.
School of Mathematics and Computer Science, Ningxia Normal University, Guyuan 756000, China
2.
School of Information Engineering, Zhengzhou Institute of Finance and Economics, Zhengzhou 450000, China

* Corresponding author
* Corresponding author

Received: July 31, 2016

Revised: January 31, 2017

Published: September 2017

The first author is supported by NSFC(11601250), the Science and Technology Pillar Program of Ningxia(KJ[2015]26(4)), the Visual Learning Young Researcher of Ningxia and the Key Program of Ningxia Normal University(NXSFZD1708), the second author is supported by the Key Program of Higher Education of Henan(16A110038, 17B110003), and the third author is supported by NSFC(11361046)

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Abstract

In this paper we study the limit cycle bifurcation of a piecewise smooth Hamiltonian system. By using the Melnikov function of piecewise smooth near-Hamiltonian systems, we obtain that at most $12n+7$ limit cycles can bifurcate from the period annulus up to the first order in $\varepsilon$.

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Mathematics Subject Classification: Primary: 34C05, 34C07; Secondary: 37G15.

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Figure 1. The closed orbits of system (1.4)$|_{\varepsilon=0}$

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Figure 2. Phase portrait of system (1.12)$|_{\varepsilon=0}$

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