Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems

Lijun  Wei; Xiang Zhang

doi:10.3934/dcds.2016.36.2803

Article Contents

2016, Volume 36, Issue 5: 2803-2825. Doi: 10.3934/dcds.2016.36.2803

This issue Previous Article Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions Next Article On the Cauchy problem of a three-component Camassa--Holm equations

Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems

Lijun Wei^1, and
Xiang Zhang^2,

1.
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240
2.
Department of Mathematics and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240

Received: November 30, 2014

Revised: July 31, 2015

Published: May 2017

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Abstract

This paper deals with the maximum number of limit cycles, which can be bifurcated from periodic orbits of planar piecewise smooth Hamiltonian systems, which are located in a neighborhood of a generalized homoclinic loop with a nilpotent saddle on a switch line. First we present asymptotic expressions of the Melnikov functions near the loop. Then using these expressions we study the number of limit cycles which are bifurcated from the periodic orbits near the homoclinic loop under small perturbations. Finally we provide two concrete examples showing applications of our main results.

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

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