Studies On The Melnikov Method And Periodic Solutions For A Class Of Planar Hybrid Autonomous Systems

Studies on dynamics for non-smooth systems are one of the leading fields, which have caused widely concerns. The objective of this paper is to develop the Melnikov methods, which can be applied to study the existence of the periodic solutions for a class of three-piecewise planar hybrid autonomous systems.Firstly, we suppose the planar is separated by two straight lines into three zones and the dynamics in each domain is governed by a smooth system. When an orbit reaches the separation lines, then a reset map describing an impacting rule applies instantaneously before the orbit enters into another domain. We assume that the unperturbed system has a continuum of periodic orbits transversally crossing the separation lines. Then, we want to study the persistence of the periodic orbits under an autonomous perturbation and the reset map. To achieve this objective, we first choose four appropriate switching sections and build a Poincaré map, after that, we present a displacement function and carry on the Taylor expansion of the displacement function to the first-order in the perturbation parameter. We define the first coefficient in the expansion as the first-order Melnikov function whose zeros provide us the persistence of periodic orbits under perturbation. Finally, we study the periodic orbits of a concrete planar hybrid piecewise-smooth system by the obtained Melnikov function.Furthermore, numerical simulations are shown to verify the effectiveness of the developed Melnikov method.The main innovations of this paper are shown as follows:1. We first study the Melnikov method of a class of three-zonal planar hybrid autonomous systems, and the diversities of impact rules applied on the switching manifolds can be defined by reset maps.2. No existence of a Hamiltonian first integral is assumed in either the perturbed or unperturbed system. Also, no symmetry assumptions are required.3. According to the theoretical analysis and numerical simulation of a concrete example, we verify the effectiveness of the developed Melnikov method whichapplied for studying the existence of periodical orbits of planar hybrid autonomous systems.