Bursting oscillation of a pendulum with irrational nonlinearity

The bursting oscillation has attracted lots of attention due to its complex dynamical behaviors. However, most models are based on polynomial nonlinearity, and the irrational nonlinearity has not been considered by researchers. Particularly, it is a typical nonlinear model that has been widely utilized in engineering fields, so it is urgent to investigate the bursting behaviors of the system. To realize this task, this paper explores a novel bursting oscillation of a pendulum with irrational nonlinearity. Three typical cases of excitation patterns are employed to investigate the complicated bursting behaviors. The mechanism of slow bursting response and the stability of equilibrium points are analyzed via the multiple time scale dynamics method. The three-parameter, two-parameter, and one-parameter bifurcations are proposed. Four new types of bifurcations, namely, forked-, spindle-, saddle- and camelback-shaped bifurcations are observed. The bursting trajectory and jumping characteristics are examined by employing the transformed phase. Finally, the multi-valued phenomenon is performed via the basin of attraction.