Pitchfork-bifurcation-delay-induced bursting patterns with complex structures in a parametrically driven Jerk circuit systemProject supported by the National Key Basic Research Program of China (Grant No.2015CB057400) and the National Natural Science Foundation of China (Grant No.11672201)

This paper aims to present the novel routes to periodic and chaotic bursting oscillations, i.e. the different routes via delayed pitchfork bifurcations, a cascade of inverse period doubling bifurcations, Hopf bifurcations and homoclinic orbits, based on the parametrically driven Jerk circuit system. Firstly, by calculating the corresponding characteristic polynomial, we obtain the stabilities of different attractors and the critical values related to different bifurcations. Moreover, the transition mechanisms among different stable attractors have been revealed, including homoclinic connection and chaotic attractors. Secondly, based on the analysis of stabilities, bifurcations and transitions among different stable attractors, we investigate the mechanisms of different bursting oscillations. A distinct delayed supercritical pitchfork bifurcation is observed when the slow-varying parameter passes through the supercritical pitchfork bifurcation point periodically. We see that the delayed behavior may terminate at different parameter areas, which leads to different types of bursting oscillations. In particular, two novel chaotic bursting patterns, the 'delayed sup-pitchfork/a cascade of inverse flip/supHopf' chaotic bursting and the 'delayed sup-pitchfork/a cascade of inverse flip/homoclinic connection/a cascade of inverse flip/supHopf' chaotic bursting, are revealed. Our research enriches the routes to bursting oscillations and deepens the understanding of bursting phenomena. Finally, the accuracy of our study is verified by the numerical simulations.