Analysis of high-order singly nested mixed-mode oscillation bifurcations

In previous works (Inaba and Kousaka, 2020; Inaba and Tsubone, 2020; Kato et al. 2022; Inaba et al., 2023), significant bifurcation structures referred to as nested mixed-mode oscillations (MMOs) were discovered in the forced Bonhoeffer-van der Pol (BVP) oscillator. Since the experimental discovery of MMOs, unnested MMO-incrementing bifurcations (MMOIBs) have been observed in numerous systems, including the Belousov–Zhabotinsky reaction, chlorite–thiosulfate reaction, and hydrodynamic and electrochemical systems; in such systems, daughter oscillations [L1s1,L2s2×m] can be observed for successive values of m. In this paper, we investigate high-order singly nested MMOs of the form [C0,m,D0,m×p] for increasing values of m, i.e., C0,m is followed by D0,m repeated p times, where C0,m≡[C0,D0×m], D0,m≡[C0,D0×(m+1)], and C0 and D0 are adjacent fundamental MMOs, e.g., C0=1s and D0=1s+1. This work extends our previous analysis that was conducted on the lowest-order case (m=1), and we demonstrate that the high-order (m≥2) singly nested MMOs can indeed occur. We investigate the cases of m=2,3,4, and 10, although we note that the regions in which nested MMOs can be observed become increasingly narrow for larger values of m. These bifurcations are confirmed via one-parameter bifurcation diagrams, time-series waveforms, and first return plots. Furthermore, we present notable similarities between the Nagumo–Sato piecewise-linear discontinuous one-dimensional map and the forced BVP oscillator.