Analysis of Successive Doubly Nested Mixed-Mode Oscillations

In previous works [Inaba & Kousaka, 2020; Inaba & Tsubone, 2020; Inaba et al., 2023], significant bifurcation structures referred to as nested Mixed-Mode Oscillations (MMOs) were found to be present in forced Bonhoeffer–van der Pol (BVP) oscillators. It is well known that unnested Mixed-Mode Oscillation-Incrementing Bifurcations (MMOIBs) can generate $[A_0,B_0\times m]$ oscillations (i.e. $A_0$ followed by $B_0$ repeated $m$ times) for successive values of $m$, where $A_0$ and $B_0$ are adjacent fundamental simple MMOs, e.g. $A_0=1^s$ and $B_0=1^{s+1}$, where $s$ is an integer. Furthermore, it has been confirmed that MMOIBs can generate nested MMOs. Let two adjacent unnested MMOIB-generated MMOs be denoted $A_1(=[A_0,B_0\times m])$ and $B_1(=[A_0,B_0\times (m+1)])$. Then, singly nested MMOIBs can generate $[A_1,B_1\times p]$ for successive values of $p$, i.e. $A_1$ followed by $B_1$ repeated $p$ times, between the $A_1$- and $B_1$-generating regions. The sequential generation of singly nested MMOs has been investigated in detail in previous work [Ito et al., 2021]. Nested MMOs can, however, be at least doubly nested. In this study, we investigate doubly nested MMOs considering a constrained nonautonomous BVP oscillator containing an idealized diode. Based on the observed dynamics of this system, Poincaré return maps are rigorously constructed in one dimension. Therefore, we can solve the successive saddle-node bifurcations using a nested (i.e. double-loop) bisection method. We track 60 successive doubly nested MMOIBs and we do not rule out the possibility that the 58 scaling constants corresponding to the MMOIB intervals converge to unity. We note that because we solve the bifurcation equation avoiding the use of a method that requires the careful selection of the initial conditions (e.g. the Newton–Raphson), we can accurately track the saddle-node bifurcations without missing any doubly nested MMO sequences.