Analysis of High-Order Nested Mixed-Mode Oscillations in a Nonautonomous Bonhoeffer–van der Pol Oscillator

Mixed-mode Oscillations (MMOs), which are characterized by alternating patterns of comparatively small- and large-amplitude oscillations, can be observed in many nonlinear systems. Understanding MMOs is essential to research in the dynamics of chemical reactions, electronic circuits, and biological systems. In many studies, a certain type of MMO bifurcations, referred to as MMO-incrementing bifurcations, has been observed. These bifurcations exhibit a repeated motif, where two basic oscillations switch in a regular way. However, recent studies have shown that novel, more complex MMO structures exist. In particular, a new class of MMO patterns has been discovered in periodically forced systems, such as the Bonhoeffer–van der Pol (BVP) and van der Pol oscillators; these oscillations have been denoted as nested MMOs. Nested MMOs are distinct from previously known MMOs because they show a hierarchical or nested structure. In this nested structure, a sequence of MMOs appears inside another repeating sequence, forming a two-level pattern. The motivation of this study is to understand how these nested MMO structures form and how they are related to the more well-known unnested MMOs; and for this purpose, we consider a nonautonomous BVP oscillator. The second-order nested MMOs observed in this work are confirmed to occur more than 20 times in succession in the nonautonomous BVP dynamics, and it is observed that the value of the scaling constants that describe the dynamics of the system, which are defined in a similar manner to Feigenbaum’s constant, may converge to a value in the vicinity of unity. This discovery adds further depth to the theory of oscillatory dynamics and opens new directions for research in nonlinear science.