The multi-bifurcation-cascade-induced periodic and chaotic bursting oscillations in a memcapacitive system

Abstract This paper aims to explore a novel route from the periodic bursting oscillations to the chaotic bursting oscillations in an externally excited memcapacitive system. In particular, these periodic bursting oscillations and chaotic bursting oscillations are induced by the multi-bifurcation cascade. Based on the fast-slow analysis method and two-parameter bifurcation analysis, it can be noticed that the existence of different bifurcation structures can induce a series of complex dynamic behaviors, which can be summarized as bursting oscillations induced by the multi-bifurcation cascade. When the system parameters change through different parameter areas, the dynamic behaviors driven by the multi-bifurcation cascade successively experience ‘period-1’ bursting, ‘intermittent period-doubling’ bursting, ‘period-2’ bursting, ‘period-2/period-4/chaos’ intermittent chaotic bursting induced by period-doubling transition, ‘period-4/chaos’ intermittent chaos bursting, ‘period-doubling degradation’ bursting, ‘period-2/chaos’ intermittent chaos bursting, ‘point/chaos’ intermittent chaos bursting, ‘period-2/chaos’ intermittent chaos bursting and chaos. These complicated dynamic behaviors constitute the important links from the periodic bursting to the chaotic bursting. We find that the period-4 bursting can degenerate into the period-2 bursting and the system directly transition from the period-2 bursting to chaos. We consider this dynamic behavior as the period-doubling degradation. In addition, we observe that the period-doubling bifurcations can cause the generation of the chaotic attractors, and the combination of the period-doubling bifurcations and inverse-period-doubling bifurcations can lead to the intermittent chaos. This work combines the complex structures such as period-doubling bifurcation, inverse period-doubling bifurcation and chaos with the traditional bursting pattern definitions, proposing a series of different bursting oscillation patterns. Our research enriches the possible routes to bursting oscillations and deepen the understanding of the bursting oscillations deeply.