Bifurcation and chaotic threshold of Duffing system with jump discontinuities

Abstract

Like for the smooth system, it is important to determine a criteria for bifurcation of the non-smooth system with jump discontinuities. Previously, the criteria have been constructed in some non-smooth systems with jump discontinuities which are the endpoints of interval. The research on bifurcation for non-smooth system with jump discontinuities, which are the interior points of interval, seems to be a new area. We construct the Duffing system with jump discontinuities in open interval. Bifurcation diagram of the unperturbed system is detected and Hamilton phase diagrams are simulated using Matlab. The jump discontinuities for the non-smooth Homoclinic orbit lead to a barrier for conventional nonlinear techniques to obtain the criteria for chaotic motion. Traditionally, these non-smooth factors were considered term by term. We will give a reasonable compromise based on all of characteristics of the non-smooth homoclinic orbits with the jump discontinuities. The extended Melnikov function is explicitly detected to judge the stable and unstable manifolds whether intersect transversally at any position of trajectory under the perturbation of damping and external forcing. It is worthwhile noting that the result reveals the effects of the non-smooth restoring force on the behaviors of nonlinear dynamical systems. The efficiency of the theoretical results is verified by the phase portraits, Poincare surface of section, Largest Lyapnnov exponents diagram and bifurcation diagram.