Bifurcation Structures of Nested Mixed-Mode Oscillations

In recently published work [Inaba & Kousaka, 2020a; Inaba & Tsubone, 2020b], we discovered significant mixed-mode oscillation (MMO) bifurcation structures in which MMOs are nested. Simple mixed-mode oscillation-incrementing bifurcations (MMOIBs) are known to generate [A(0),B-0 x n] oscillations for successive n between regions of A(0)- and B-0-oscillations, where A(0) and B-0 are adjacent simple MMOs, e.g. A(0) = 1s and B-0 = 1(s+1), where s is an integer. MMOIBs are universal phenomena of evidently strong order and have been studied extensively in chemistry, physics, and engineering. Nested MMOIBs are phenomena that are more complex, but have an even stronger order, generating chaotic MMO windows that include sequences [A(1),B-1 x n] for successive n, where A(1) and B-1 are adjacent MMOIB-generated MMOs, i.e. A(1) = [A(0),B-0 x m] and B1 = [A(0),B-0 x (m + 1)] for integer m. Herein, we investigate the bifurcation structures of nested MMOIB-generated MMOs exhibited by a classical forced Bonhoeffer-van der Pol oscillator. We use numerical methods to prepare two- and one-parameter bifurcation diagrams of the system with m = 1, 2, and 3 for successive n for the case s = 2. Our analysis suggests that nested MMOs could be widely observed and are clearly ordered phenomena. We then define the first return maps for nested MMOs, which elucidate the appearance of successively nested MMOIBs.