Impact oscillators with homoclinic orbit tangent to the wall

Homoclinic bifurcation for a nonlinear inverted pendulum impacting between two rigid walls under external periodic excitation is analyzed under the hypothesis that the unperturbed system has a homoclinc orbit tangent to the wall. Consequently, the impact surface cannot be chosen as the Poincaré section to measure the distance between the perturbed stable and unstable manifolds. Furthermore, compared to the case that the unperturbed homoclinic orbit intersects the wall transversally, more cases are involved as the parameters vary. Thus the analysis of the homoclinic orbit tangent to the wall is much more difficult. In this paper, by using a method of Melnikov type, we derive sufficient conditions under which the perturbed stable and unstable manifolds intersect transversally. As an application, an impact oscillator of Duffing type is studied in detail.