Quasi-periodic solutions and homoclinic bifurcation in an impact inverted pendulum

We investigate a nonlinear inverted pendulum impacting between two rigid walls under external periodic excitation. Based on KAM theory, we prove that there are three regions (corresponding to different energies) occupied by quasi-periodic solutions in phase space when the excitation is small. Moreover, rotational motion maintained perturbation gets larger. The existence of subharmonic obtained Aubry–Mather theory and boundedness all followed fact exist abundant invariant tori near infinity. To study homoclinic bifurcation this system, present numerical method compute discontinuous manifolds accurately, which provides useful tool for effect impacts.