Global bifurcations of a smooth and discontinuous double-winged oscillator

In this paper, the global bifurcations of a double-winged oscillator with geometrical constraint under a nonlinear perturbation are investigated using the subharmonic Melnikov function for both perfect elastic impact case and imperfect elastic impact case. Stability analysis of discontinuous singularities is carried out, upon which the discontinuous Hopf bifurcation is discovered leading to a pair of discontinuous limit cycles which always exists when the system is dissipated by imperfect elastic impact. Also, an extended method of the subharmonic Melnikov function is introduced to detect the limit cycles for imperfect elastic impact case, which helps us to obtain the global bifurcation sets including smooth and discontinuous Hopf bifurcation, homoclinic and heteroclinic bifurcation, closed orbit bifurcation, and also the subcritical pitchfork bifurcation. The multiple bifurcation sets are divided into some subsets including bifurcation curves and intersection points depending on the geometrical parameter β and perturbation parameter α. Phase structures corresponding to each subset and each area are constructed.