Sliding bursting oscillations related to transcritical bifurcation delay in an excited vector field with frequency switching

The purpose of this paper is to report sliding bursting dynamics related to transcritical bifurcation delay, a common route to bursting dynamics. As an example, a simple nonlinear oscillator with a slow parametric excitation is considered, in which bursting oscillations resulted from transcritical bifurcation delay can be observed. Typically, such oscillations exhibit prolonged rest phases, characterized by the oscillations that moves along the unstable origin of the system. We show that interesting dynamical characteristics can be observed in the bursting oscillations if the excitation frequency switches between two different values. In particular, additional rest phases, i.e., the newly created non-zero sliding behaviors, characterized by that the trajectory slides along the frequency switching threshold for some time, can be exhibited in the presence of frequency switching. This forms the so-called sliding bursting oscillations. Then, we investigate the dynamical mechanisms of these oscillations and explore their transitions in relation to the variation of frequency switching threshold. As a result, several different patterns of sliding bursting oscillations are obtained. In particular, we find that sliding bursting oscillations may diverge to infinity for some threshold windows, and this can be understood by the analysis of attracting basins.