Nested mixed-mode oscillations, Part III: Comparison of bifurcation structures between a driven Bonhoeffer–van der Pol oscillator and Nagumo–Sato piecewise-linear discontinuous one-dimensional map

• Published in: Physica. D Vol. 446; p. 133667
• Main Authors: Inaba, Naohiko, Tsubone, Tadashi, Ito, Hidetaka, Okazaki, Hideaki, Yoshinaga, Tetsuya
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.04.2023
• Subjects: Driven Bonhoeffer–van der Pol oscillator; Mixed-mode oscillation-incrementing bifurcations; Nagumo–Sato map; Nested mixed-mode oscillations; Nested period-adding bifurcations; Nagumo–Sato map; Mixed-mode oscillation-incrementing bifurcations; Driven Bonhoeffer–van der Pol oscillator; Nested period-adding bifurcations; Nested mixed-mode oscillations
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2023.133667

In our previous studies (Inaba and Kousaka, (2020); Inaba and Tsubone, (2020)), we discovered bifurcation structures represented by nested mixed-mode oscillations (MMOs) generated by a driven Bonhoeffer–van der Pol (BVP) oscillator. BVP oscillators are equivalent to FitzHugh–Nagumo models and have been a subject of intense research for the last six decades. In this study, we consider the case in which the diode included in a driven BVP oscillator is assumed to operate as an ideal switch. In this case, Poincaré return maps can be rigorously constructed one-dimensionally, which consist of two downward convex branches. We also consider the Poincaré return map that is approximated as a two-segment piecewise-linear discontinuous one-dimensional map. Such a piecewise-linear map was proposed by Nagumo and Sato and generates nested period-adding bifurcations. We show that un-nested, singly, and doubly nested MMO-incrementing bifurcations generated by the driven BVP oscillator coincide with one of the possible un-nested, singly, and doubly nested period-adding bifurcations, respectively, generated with the Nagumo–Sato map. •Bonhoeffer–van der Pol (BVP) oscillators exhibit FitzHugh–Nagumo dynamics.•We discuss nested mixed-mode oscillations (MMOs) generated by a BVP oscillator.•The oscillator with a diode has a canard without a head in the absence of perturbation.•Nested MMOs are analyzed with one-dimensional Poincaré return maps.•The Nagumo–Sato discontinuous one-dimensional map explains nested MMOs.