The discontinuous limit case of an archetypal oscillator with a constant excitation and van der Pol damping

Abstract

We investigate global dynamics of the discontinuous limit case of an archetypal oscillator with a constant excitation that is a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load. The discontinuous dynamical system can exhibit three limit cycles and rich nonlinear phenomena, including boundary equilibrium bifurcation, Hopf bifurcation, grazing bifurcation, pseudo saddle–node loop bifurcation and double crossing limit cycle bifurcation. Because of destroyed symmetry for this system, it exhibits more complex and abundant dynamics than the smooth oscillator with van der Pol damping and can appear new dynamical phenomena, such as coexistence of a crossing limit cycle and a pseudo homoclinic loop, coexistence of two crossing limit cycles and the existence of pseudo saddle–node loops. The bifurcation diagram of the discontinuous system is shown in parameter space and all global phase portraits in the Poincaré disk are sketched in the phase plane.