Complex mixed-mode oscillations in a Bonhoeffer–van der Pol oscillator under weak periodic perturbation

Abstract

In this paper, we elucidate the extremely complicated bifurcation structure of a weakly driven relaxation oscillator by focusing on chaos, and notably, on complex mixed-mode oscillations (MMOs) generated in a simple dynamical model. Our model uses the Bonhoeffer–van der Pol (BVP) oscillator subjected to a weak periodic perturbation near a subcritical Andronov–Hopf bifurcation (AHB). The mechanisms underlying the chaotic dynamics can be explained using an approximate one-dimensional map. The MMOs that appear in this forced dynamical model may be more sophisticated than those appearing in three-variable slow–fast autonomous dynamics because the approximate one-dimensional mapping of the dynamics used herein is a circle map, whereas the one-dimensional first-return map that is derived from the three-variable slow–fast autonomous dynamics is usually a unimodal map. In this study, we generate novel bifurcations such as an MMO-incrementing bifurcation (MMOIB) and intermittently chaotic MMOs. MMOIBs trigger an MMO sequence that, upon varying a parameter, is followed by another type of MMO sequence. By constructing a two-parameter bifurcation diagram, we confirmed that MMOIBs occur successively. According to our numerical results, MMOIBs are often observed between two neighboring MMOs. Numerically, MMOIBs may occur as many times as desired. We also derive the universal constant of the associated successive MMOIBs. The existence of the universal constant suggests that MMOIBs could occur infinitely many times. Furthermore, intermittently chaotic MMOs appear in this dynamical circuit. The intermittently chaotic MMOs relate to a type of intermittent chaos that resembles MMOs at first glance, but includes rare bursts over a long time interval. Complex intermittently chaotic MMOs of various types are observed, and we clarify that the intermittently chaotic MMOs are generated by crisis-induced intermittency.