Nested mixed-mode oscillations, Part III: Comparison of bifurcation structures between a driven Bonhoeffer–van der Pol oscillator and Nagumo–Sato piecewise-linear discontinuous one-dimensional map

Abstract

In our previous studies (Inaba and Kousaka, (2020); Inaba and Tsubone, (2020)), we discovered bifurcation structures represented by nested mixed-mode oscillations (MMOs) generated by a driven Bonhoeffer–van der Pol (BVP) oscillator. BVP oscillators are equivalent to FitzHugh–Nagumo models and have been a subject of intense research for the last six decades. In this study, we consider the case in which the diode included in a driven BVP oscillator is assumed to operate as an ideal switch. In this case, Poincaré return maps can be rigorously constructed one-dimensionally, which consist of two downward convex branches. We also consider the Poincaré return map that is approximated as a two-segment piecewise-linear discontinuous one-dimensional map. Such a piecewise-linear map was proposed by Nagumo and Sato and generates nested period-adding bifurcations. We show that un-nested, singly, and doubly nested MMO-incrementing bifurcations generated by the driven BVP oscillator coincide with one of the possible un-nested, singly, and doubly nested period-adding bifurcations, respectively, generated with the Nagumo–Sato map.