Bursting patterns with complex structures in a parametrically and externally excited Jerk circuit system

This paper aims to report complex bursting patterns in a parametrically and externally excited Jerk circuit system. Typically, the common compound bursting with two clusters in each period can be observed in this system. By the fast-slow analysis, we find that the active states related to this type of bursting are created by supercritical Hopf bifurcations. Interestingly, with the increase of parametric excitation amplitude, the common bursting pattern evolves into the ones that display complex structures. We show that the parametric excitation amplitude plays a vital role in influencing the dynamical behaviors, which is apparent from the two-parameters analysis. Then, a rich variety of dynamical phenomena including period-doubling and inverse period-doubling cascades in orbits of period 2, 4, and 8 can be observed with the increase of parametric excitation amplitude, thereby leading to complex bursting patterns with multiple-periodically active states. Besides, as a result of the period-doubling cascades, chaotic attractors are eventually discovered, which can be corroborated by the maximal Lyapunov exponent. In particular, periodic windows with period-doubling and inverse period-doubling cascades also emerge in the chaotic region when the parametric excitation amplitude increases to higher values. These give rise to the complex bursting patterns with chaotically active states. Based on this, two types of bursting patterns with complex structures, i.e., bursting patterns with multiple-periodically active states and bursting patterns with chaotically active states are revealed. Our results enrich the routes to complex bursting and deepen the understanding of bursting patterns with complex structures.