Successive nested mixed-mode oscillations

Abstract

In a previous study [Inaba et al., Physica D (2020), web on-line], we discovered nested mixed-mode oscillations (MMOs) generated by a Bonhoeffer-van der Pol (BVP) oscillator under weak periodic perturbations near a subcritical Hopf bifurcaton point. The dynamics of BVP oscillators are equivalent to FitzHugh-Nagumo dynamics and have been studied extensively for more than five decades. In this study, we focus on the singly nested MMOs that occur between the 1⁴- and 1⁵-generating regions in a piecewise-smooth BVP oscillator with an idealized diode where 1^s indicates alternating time-series waveforms that consist of a large amplitude oscillation followed by s small peaks, and we confirm 400 nested mixed-mode oscillation-incrementing bifurcations (MMOIBs). Our numerical results suggest that the universal constant converges to one, which was predicted because MMOIBs increment and terminate toward an MMO increment-terminating tangent bifurcation point and the gradient of the tangent points is one.