Homoclinic bifurcations and chaotic dynamics for a piecewise linear system under a periodic excitation and a viscous damping

Abstract

In this paper, homoclinic bifurcations and chaotic dynamics of a piecewise linear system subjected to a periodic excitation and a viscous damping are investigated by the Melnikov analysis for nonsmooth systems in detail. The piecewise linear system can be seen as a simple linear feedback control system with dead zone and saturation constrains. The unperturbed system is a piecewise linear Hamiltonian system, which contains two parameters and exhibits quintuple well characteristic. The discontinuous unperturbed system, which is obtained by reducing the two parameters to zero, has saddle-like singularity and homoclinic-like orbit. Analytical expressions for the unperturbed homoclinic and heteroclinic orbits are derived by using Hamiltonian function for the piecewise linear system. The Melnikov analysis for nonsmooth planar systems is first described briefly, and the theorem for homoclinic bifurcations for the nonsmooth planar systems is also obtained and then employed to detect the homoclinic and heteroclinic tangency under the perturbation of a viscous damping and a periodic excitation. Finally, the chaotic attractors and the bifurcation diagrams are presented to show the bifurcations and chaotic dynamics of the piecewise linear system.