Periodic impact motions at resonance of a particle bouncing on spheres and cylinders (Q2363897)

The author considers the dynamics of a particle bouncing against a sphere, without friction, while being attracted towards its center by a force which, at large distances, has a linear behaviour. More precisely, the nonlinearity is assumed to stay between two asymptotes of the curves of the Fučík spectrum, a kind of nonresonance condition. However, a closer approach to resonance is considered, as well, by assuming Landesman-Lazer type of conditions at both sides. This type of situation is sometimes referred to as ``double resonance''. The main result states that, if the nonlinearity is periodic in time, there exist periodic solutions which rotate around the sphere a given number of times, while bouncing periodically on it. The proof relies on degree theory, and on careful estimates in the phase plane. Similar results are stated also in the case of the particle being assumed to bounce on a cylinder, instead of a sphere.