Bifurcations and mixed mode oscillations in a bi-stable plasma model with slow parametric excitation

Abstract

In this study, considering a bi-stable plasma model with slow parametric excitation, the bifurcation of periodic and chaotic responses as well as the resulting fast–slow motions is discussed analytically and numerically. For a nonautonomous fast sub-system, the generalized harmonic balancing method is utilized to obtain an averaged system. Bifurcation analysis about the averaged system shows that the critical manifolds form a S-shape structure. Meanwhile, supercritical and subcritical period doubling (PD) occurs on the upper branch simultaneously. As the frequency of the external excitation changes, bifurcation points on the limit cycle manifolds can present different relative locations. Moreover, an additional bi-stable structure induced by Cusp bifurcation emanates from the upper branch. On the other hand, the existence of a chaotic attractor and the corresponding boundary crisis phenomenon are verified using the Melnikov method and the basin of attraction. The structures of the numerical bifurcation diagram show good agreements with the analytical results. Considering two cases of low-frequency excitation, the corresponding fast–slow dynamics are discussed. It is found that, when the fast–slow flow passing the subcritical PD point, a low frequency with different magnitudes will lead to two patterns of bifurcation delay, i.e., the typical one and the excessive delay, which suppress the PD. As for the boundary crisis point, the slow passage effects show no distinct influence. Thus, three transition mechanisms based on two cases of the bifurcation structure are explained, including “fold of cycle–fold of cycle” type, “fold of cycle–delayed subcritical PD” type, and “fold of cycle–boundary crisis” type.