Bifurcation analysis of mixed-mode oscillations and Farey trees in an extended Bonhoeffer-van der Pol oscillator

The two-variable Bonhoeffer-van der Pol oscillator, which is equivalent to the FitzHugh-Nagumo model, can be represented by a natural circuit, i.e., a circuit consisting only of simple two-terminal elements. An extended Bonhoeffer-van der Pol (BVP) oscillator is a circuit extended to a three-variable system from a two-variable BVP oscillator; this is obtained by adding an inductor-resistor branch to the original BVP oscillator. The extended BVP oscillator is known to generate mixed-mode oscillations. In this study, we classify the bifurcations of the simple sequences and their daughters, which constitute asymmetric Farey trees. Because the extended BVP oscillator is an extremely simple circuit, the parents-daughter processes can be quite precisely explained. We confirm that simple sequences 1(s) (s >= 0) are born via saddle-node bifurcations and can be basic parents, which correspond to each stable fixed point of a Poincare return map, where 1(s) represents one large excursion followed by a number s of small peaks. The two basic parents 1(s) and 1(s+1) generate daughters [1(s+1), 1(s) x n] sequentially for the successive values of n via mixed-mode oscillation-incrementing bifurcations. The terms [1(s+1), 1(s) x n] indicate that 1(s+1) is followed by 1(s) n-times. The daughters are born in a similar fashion to period-adding bifurcations generated by the circle map, and the parents-daughter processes satisfy Farey arithmetic and are terminated by a saddle-node bifurcation through which 1(s) appears. We create multiple two-parameter bifurcation diagrams and confirm that these phenomena are stably observed in these diagrams over broad ranges, i.e., complex bifurcations, such as codimension-two bifurcations or cusps, do not appear in the diagrams. Furthermore, the theoretical results are confirmed using laboratory measurements and experiments. (C) 2022 Elsevier B.V. All rights reserved.

Automated summary

This study investigates the two-variable Bonhoeffer-van der Pol oscillator and its extended version, revealing that the extended BVP oscillator generates mixed-mode oscillations and classifies the simple sequences and their offspring. By explaining the parent-daughter processes, it is found that various phenomena can be stably observed.