Two fast/slow decompositions as well as period-adding sequences in a generalized Bonhoeffer-van der Pol electronic circuit

Abstract

Bursting dynamics in a generalized Bonhoeffer-van der Pol electronic circuit can reflect multi-timescale characteristics. We use two fast-slow decompositions for the study of the mixed mode oscillations and period-adding sequences observed in this GBVP system. Assuming time-scale ratio clarified by parameter ε, with one fast and two slow variables, a singular periodic orbit can be constructed by solving the desingularized system for the flow on the upper and lower sheets of three-dimensional critical manifold. Such dynamics due to the folded singularity and S-shaped surface. The variation of the parameter values lead to more spikes of the bursting as the upper attracting and center repelling sheets intersect and twist near the folded node. On the other hand, if we choose scale ratio as the other parameter c, two-fast/one-slow analysis could be used. Here, one treats the slow variable as a parameter of the fast subsystem, and studies the bifurcation structure of this subsystem including Bogdanov-Takens (BT), generalized Hopf (GH) and cusp bifurcations. Characteristic features of this case are subcritical Hopf bifurcation and slow passage in close distance to the jump. Furthermore, the cascades of period-adding sequences, resulting in chaotic bursting are observed and discussed in detail.