Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation

A class of periodic motions of an inverted pendulum with rigid lateral barriers is analysed under the hypothesis that the system is forced by impulsed periodic excitation. Due to the piece-wise linear nature of the problem, the existence and the stability of the cycles are determined analytically. It is found that they depend on both classical (saddle-node and period-doubling) and non-classical bifurcations, the latter involving a `synchronizationʹ between impulses and impacts which leads to the sudden disappearing of the orbits. Attention is paid to the physical interpretation of these bifurcations, and to the determination of analytical criteria for their occurrence. We study how the relative position (with respect to the excitation amplitude) of the local bifurcations determines the system response and the bifurcation scenario. Symmetric and unsymmetric excitations are considered and the regions of stability of the periodic solutions are analytically determined. Finally, a comparison with the case of harmonic excitation is presented showing both analogies and differences, and highlighting how the impulsed excitation allows to obtain stable periodic responses at higher values of the excitation amplitude.