Semi-Analytical Periodic Motions and Bifurcations in a Periodically Impulsive, Damped Pendulum

In this paper, a computational method is developed for the bifurcation dynamics of periodic motions in impulsive nonlinear dynamical systems. Such a method based on the implicit mapping method is to discretize the nonlinear dynamical systems in the time interval of two impulses. The impulsive pendulum is studied as an example herein. Using specific mapping structures, periodic motions in the impulsive dynamical systems are determined, and the corresponding stability and bifurcation analysis of the periodic motions are carried out. The impulsive homoclinic orbits related to the different periodic motions are achieved, and the homoclinic bifurcations of periodic motions are determined. The sampled periodic motions and impulsive homoclinic orbits are illustrated for a better understanding of the bifurcation dynamics of impulsive nonlinear systems. Such a method can be applied to other impulsive nonlinear systems.