Chaotic threshold for non-smooth system with multiple impulse effect

Abstract

An inverted pendulum with multiple impulse effect is constructed to detect chaotic threshold for a class of impulsive differential system. These impulsive excitations do not necessarily act on the equilibria of the non-smooth system, which leads to a barrier for the conventional nonlinear techniques. How to solve this barrier is an issue on the top of our agenda. Normally, these non-smooth factors were considered term by term. Using Hamiltonian function and principle perturbation theory, a reasonable compromise is applied to detect the extended Melnikov function based on the characteristics of the non-smooth homoclinic orbits with multiple jump discontinuities. It is a critical step that the perturbation solutions are examined from two points of view in order to obtain their small-perturbation expansions. Furthermore, the Melnikov function is employed to obtain the criteria for chaotic motion, which shows the effects of these impulsive excitations, the perturbation of damping and external forcing on the behaviors of the non-smooth dynamical system. The efficiency of the criteria for chaotic motion obtained in this paper is verified by bifurcation diagram, phase portraits and Poincare surface of section.