Bursting oscillations and bifurcation mechanism in memristor-based Shimizu-Morioka system with two time scales

Bursting oscillators have received great attention in recent years, however, the research on this issue associated with memristive systems has been rarely reported. In this paper, bursting oscillations and bifurcation mechanism in a memristor-based Shimizu-Morioka system are investigated when an order gap exists between the excitation frequency and the natural frequency. Firstly, the bifurcation properties of the fast system are exploited by considering the periodic excitation as a slow-varying parameter. And the stability of different attractors and the critical values of different bifurcations are obtained. Secondly, Complex bursting oscillators are revealed when the slow-varying parameter passes through these critical values. The corresponding bifurcation mechanism, namely, symmetric Fold/Fold, symmetric compound Fold/Fold-delayed supHopf/supHopf, symmetric compound subHopf/subHopf-supHopf/supHopf, symmetric subHopf/subHopf, supHopf/saddle on limit cycle, symmetric delayed supHopf/delayed supHopf, symmetric delay supHopf-supHopf/supHopf are analyzed by the transformed phase portraits, the time series, and the phase portraits. Furthermore, the effect of the excitation frequency on the symmetric Fold/Fold bursting is also revealed. Finally, some numerical and circuit simulation results are provided to verify the validity of the study. (C) 2019 Elsevier Ltd. All rights reserved.