Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 2803 - 2825, Volume 36, Issue 5, May 2016      doi:10.3934/dcds.2016.36.2803

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Lijun Wei - Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China (email)
Xiang Zhang - Department of Mathematics and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China (email)

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.

Keywords:  Piecewise smooth system, generalized homoclinic loop, Melnikov function, limit cycle bifurcation, nilpotent saddle.
Mathematics Subject Classification:  Primary: 37G15, 34C07; Secondary: 34C05.

Received: December 2014;      Revised: August 2015;      Available Online: October 2015.