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 periodic excitation is small. Moreover, the rotational quasi-periodic motion is maintained when the perturbation gets larger. The existence of subharmonic periodic solutions is obtained by the Aubry–Mather theory and the boundedness of all solutions is followed by the fact that there exist abundant invariant tori near infinity. To study the homoclinic bifurcation of this system, we present a numerical method to compute the discontinuous invariant manifolds accurately, which provides a useful tool for the study of invariant manifolds under the effect of impacts.
The authors are grateful to the anonymous referees for a careful reading and suggestions that led to an improvement of the paper. This work is supported by the National Natural Science Foundation of China (12172306,11732014).
- KAM theory
- Quasi-periodic solution
- Impact system
- Computation of discontinuous invariant manifold