Mixed-mode oscillation-incrementing bifurcations and a devil’s staircase from a nonautonomous, constrained Bonhoeffer–van der Pol oscillator

In this study, we analyze mixed-mode oscillation-incrementing bifurcations (MMOIBs) generated in the nonautonomous, constrained Bonhoeffer–van der Pol oscillator proposed by Kousaka et al. [Physica D 353–354, 48 (2017)]. Specifically, we investigate MMOIBs occurring in the |1⁴|–|1⁵| and |1¹|–|1²| regions. These two kinds of MMOIBs exhibit qualitatively different MMO-bifurcation structures. The former MMOIBs successively occur many times, while the latter exhibit finite MMOIBs. In the latter case, standard MMOIBs occur only five times, and are then followed by another type of MMOIB. However, the following MMOIBs are also only generated seven times and the solution finally settles down into a |2⁰| attractor. We clarify the exact reason for these phenomena by analyzing 1D Poincaré return maps derived from the constrained dynamics. By focusing on the initial successive MMOIBs, we create asymmetric Farey trees that occur between |1⁴| and |1⁵| by analyzing the 1D Poincaré return map. We find that there exist two sets of successive MMOIBs between |1⁴| and |1⁵|⁠. In particular, we rigorously define the MMO increment-terminating tangent bifurcations, toward which the MMOIBs accumulate and terminate. Furthermore, we uncover a nested bifurcation structure caused by MMOIBs. This occurs inside a short interval in the |1⁴|–|1⁵| region and accumulates toward another MMO increment-terminating tangent bifurcation point. These three types of successively generated MMOIBs accumulate in different ways toward the MMO increment-terminating tangent bifurcation points. We also analyze the behavior of the “firing number,” which varies with the MMOIBs. In particular, we theoretically explain why a firing number that exhibits a devil’s staircase has higher values in chaos-generating regions than in MMO-generating regions.