Hopf Bifurcations for the Recently Proposed Smooth-and-Discontinuous Oscillator

We investigate the Hopf bifurcations of the recently proposed smooth-and-discontinuous (SD) oscillator which exhibits both smooth and discontinuous dynamics depending on the value of a parameter alpha. The nonlinearity presented in this system characterizes irrationality and piecewise linearity for smooth and discontinuous cases, respectively, which could not meet the requirements of the conventional methods due to the barrier of Taylor expansion. Introducing a series of new kinds of elliptic integrals of the first and second kind to the perturbed oscillator, we obtain the Poincare-Birchoff normal forms of Hopf bifurcations for both smooth and discontinuous regimes. We also demonstrate the criteria for the occurrence of Hopf bifurcations, the stability of periodic solutions bifurcating from the equilibria and the excellent agreement between the theoretical and numerical results.