Qualitative correspondence between nested mixed-mode oscillations in FitzHugh–Nagumo dynamics and the Nagumo–Sato map

It has been established (Inaba and Kousaka, 2020; Inaba et al., 2023; Kato et al., 2025) that bifurcation structures known as nested mixed-mode oscillations (MMOs) exist in the driven Bonhoeffer–van der Pol oscillator; this oscillator is equivalent to the FitzHugh–Nagumo (FHN) model. Simulations indicate that MMOs can be at least quadruply nested, and there is no evidence indicating that they cannot be nested to an arbitrarily high degree. In this Letter, we present a correspondence between nested MMOs and the Nagumo–Sato two-segment piecewise-linear discontinuous one-dimensional map. The Nagumo–Sato map is known to generate infinitely nested period-adding bifurcations; this map exhibits two unnested, four singly, eight doubly, 16 triply, and 32 quadruply nested period-adding sequences. We demonstrate that for each nesting level, the nested MMO sequences observed in the FHN model coincide with one of the possible period-adding bifurcation sequences generated by the Nagumo–Sato map.