Nested mixed-mode oscillations in a canard-generating driven Bonhoeffer–van der Pol oscillator

In previous works (Inaba and Kousaka, 2020; Inaba and Tsubone, 2020), the existence of nested mixed-mode oscillation (MMO) bifurcation structures, generated by a driven slow–fast Bonhoeffer–van der Pol (BVP) oscillator, was confirmed. This oscillator has a stable focus and a stable relaxation oscillation caused by a subcritical Hopf bifurcation in the absence of perturbation. In this study, we assert that nested MMOs in this circuit can occur regardless of whether the oscillation is caused by a supercritical or subcritical Hopf bifurcation when no perturbation is applied, and that therefore, this phenomenon could be widespread. We show that MMOs that occur under weak periodic perturbations, generated in a classical BVP oscillator, where the oscillation is caused by a supercritical Hopf bifurcation, can be at least doubly nested. To show that nested MMOs can be at least doubly nested, we utilize first return plots, which are approximated one-dimensional maps and well explain mixed-mode oscillation-incrementing bifurcations.