Relaxation oscillations and the mechanism in a periodically excited vector field with pitchfork–Hopf bifurcation

The traditional geometric singular perturbation theory and the slow–fast analysis method cannot be directly used to explore the mechanism of the relaxation oscillations in the forced dynamical system in which the exciting frequency is far less than the natural frequency. Furthermore, higher codimensional bifurcations may result in more complicated alternations between the large-amplitude oscillations (spiking states, SPs) and small-amplitude oscillations or at rest (quiescent states, QSs) on the trajectories of the bursting oscillations, called also the mixed mode oscillations. For the purpose, here we present a method to explore the mechanism of the bursting oscillations in the excited oscillator. Without loss of generality, we apply the proposed method to explore the dynamics of the normal form of the vector field with codimension-two pitchfork–Hopf bifurcation at the origin. When the slow-varying parametric excitation is introduced, with the increase in exciting amplitude, different types of bursting oscillations can be observed, the mechanism of which can be obtained by employing the overlap of the transformed phase portraits and the equilibrium branches as well as the bifurcations of the generalized autonomous system.