The Melnikov method of heteroclinic orbits for a class of planar hybrid piecewise-smooth systems and application
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In this work, we extend the well-known Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth systems, defined in three zones separated by two switching manifolds \(x=-\alpha \) and \(x=\beta \). We suppose that the dynamic in each zone is governed by a smooth system. When a trajectory reaches the switching manifolds, then reset maps describing impacting rules on the switching manifolds will be applied instantaneously before the trajectory enters into the other zone. We also assume that the unperturbed system is a piecewise-defined continuous Hamiltonian system and possesses a pair of heteroclinic orbits transversally crossing the switching manifolds. Then, we study the persistence of the heteroclinic orbits under a non-autonomous periodic perturbation and the reset maps. In order to obtain this objective, we derive a Melnikov-type function by using the Hamiltonian function to measure the distance of the perturbed stable and unstable manifolds in this system. Finally, we employ the obtained Melnikov-type function to study the persistence of a heteroclinic cycle and complicated dynamics near the heteroclinic cycle for a concrete planar piecewise-smooth system.
KeywordsMelnikov method Planar hybrid piecewise-smooth systems Heteroclinic orbits Switching manifolds Reset maps
The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant Nos. 11472298, 11290152, 11427801, 11272063, 11472056, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB), the Natural Science Foundation of Tianjin City through Grant No. 13JCQNJC04400 and the Fundamental Research Funds for the Central Universities through Grant No. 3122013k005.
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