Mixed-mode oscillations from a constrained extended Bonhoeffer-van der Pol oscillator with a diode.

An extended Bonhoeffer-van der Pol (BVP) oscillator is a circuit that is naturally extended to a three-variable system from a two-variable BVP oscillator. A BVP oscillator is known to exhibit a canard explosion, and the extended BVP oscillator generates mixed-mode oscillations (MMOs). In this work, we considered a case study where the nonlinear conductor in the extended BVP oscillator includes an idealized diode. The idealized case corresponds to a degenerate case where one of the parameters tends to infinity, and circuit dynamics are represented using a constrained equation, and at the expense of the model's naturalness, i.e., in a case in which the solutions of the dynamics are defined only forward in time, the Poincaré return maps are constructed as one-dimensional (1D). Using these 1D return maps, we explain various phenomena, such as simple MMOs and MMO-incrementing bifurcations. In this oscillator, there exists a small amplitude oscillation, which emerges as a consequence of supercritical Hopf bifurcation, and there exists large relaxation oscillation which appears via canard explosion by changing the bifurcation parameter. Between these small and large amplitude oscillations, the MMO bifurcations exhibit asymmetric Farey trees. Furthermore, these theoretical results were verified using laboratory measurements and experiments.