Nested mixed-mode oscillations

Abstract

This paper discusses nested mixed-mode oscillations (MMOs) generated by a nonautonomousBonhoeffer–van der Pol oscillator with a diode. We use our previously-proposed constrained piecewise-smooth dynamics, where the diode is assumed to operate as a switch (Kousaka et al., 2017). This idealized model corresponds to a degenerate case in which one of the parameters tends to infinity and the circuit dynamics are represented by a one-variable piecewise nonautonomous equation, enabling one-dimensional Poincaré return maps to be rigorously defined. We use these maps to investigate the bifurcation structures between the 14- and 15-generating regions. Mixed-mode oscillation-incrementing bifurcations (MMOIBs) cause period-adding sequences, denoted by 14(15)n or, more precisely, [14,15×n]n+1. We explain the mechanism that causes the invariant intervals to alternately appear and disappear as the bifurcation parameter changes to capture each MMOIB-generated MMO. The main aim of this study is to clarify the bifurcation structures of nested MMOIBs, which generate [[14,15×1]2,[14,15×2]3×n]3n+2 MMO sequences for successive n between the [14,15×1]2- and [14,15×2]3-generating regions, and also [[14,15×2]3,[14,15×3]4×n]4n+3 MMO sequences for successive n between the [14,15×2]3- and [14,15×3]4-generating regions, suggesting that MMOIBs cause successive [[14,15×m]m+1,[14,15×(m+1)]m+2×n](m+2)n+(m+1) sequences (n=1,2,3...) between the [14,15×m]m+1- and [14,15×(m+1)]m+2-generating regions, where m is an integer. We also observe doubly nested MMOIB structures, although we find significant differences between singly and doubly nested MMOIBs. Whereas clear sequences of singly nested MMOIB-generated MMOs appear in the one-parameter bifurcation diagram, the MMO-generating regions in the doubly nested case are extremely narrow in comparison (although we do confirm that they also occur successively). The generation of singly and doubly nested MMOIB sequences can be explained by the one-dimensional maps at the MMO increment-terminating tangent bifurcation points toward which the MMOIBs accumulate.