# American Institute of Mathematical Sciences

November  2017, 16(6): 2321-2336. doi: 10.3934/cpaa.2017114

## 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

Received  August 2016 Revised  February 2017 Published  July 2017

Fund Project: 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)

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 $\varepsilon$.

Citation: Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114
##### References:

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##### References:
The closed orbits of system (1.4)$|_{\varepsilon=0}$
Phase portrait of system (1.12)$|_{\varepsilon=0}$
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