The analytical method of studying subharmonic periodic orbits for planar piecewise-smooth systems with two switching manifolds

Abstract

Piecewise linear approach is one of important methods to study nonlinear dynamics of oscillators with complex nonlinear restoring forces, so piecewise-smooth systems with multiple switching manifolds sometimes are ideal mathematical models to analyze the nonlinear dynamics of nonlinear oscillators. In this paper, inspired by the work presented by Cao et al. (Philos Trans R Soc A 366(1865):635–652, 2008) and Granados et al. (SIAM J Appl Dyn Syst 11(3):801–830, 2012), we want to formulate an analytical method for studying subharmonic periodic orbits for piecewise-smooth systems with two switching manifolds. We will define what are subharmonic orbits for this class of piecewise-smooth systems and develop a Melnikov-type analytical method to detect the existence of subharmonic orbits. In order to obtain this objective, we assume that the plane is divided into three zones by the two switching manifolds which may not be symmetric, and the dynamics in each zone is decided by a smooth system. Furthermore, we suppose that the unperturbed system is a piecewise-defined Hamiltonian system and possesses a pair of piecewise-smooth homoclinic orbits crossing the two switching manifolds transversally exactly twice and connecting origin to itself. The region outside the pair of homoclinic orbits is fully covered by periodic orbits crossing respectively the two switching manifolds transversally exactly twice. Finally, we want to study the persistence of those continuum of periodic orbits outside the pair of homoclinic orbits when a small time-periodic perturbation is considered. By choosing an appropriate Poincaré section, constructing a Poincaré map and applying perturbation techniques, we obtain the Melnikov function of subharmonic orbits for the class of planar piecewise-smooth systems and employ it to study the existence of subharmonic periodic motions for a piecewise-linear oscillator. Numerical simulations are also shown the effectiveness of the analytical method to detect the parameters and initial conditions for the existence of subharmonic orbits in piecewise-smooth oscillators.