On the boundedness of solutions to a nonlinear singular oscillator (Q845740)

The authors consider a second order scalar differential equation of the form \[ x''+V'(x)=p(t), \] where \(p\) is periodic and the potential \(V\) has a strong singularity of repulsive type. Sufficient conditions are given for boundedness of all solutions, as well as for existence of Aubry-Mather sets. In the proofs, the Hamiltonian is written in area-angle variables and then careful estimates on the Poincaré map are crafted in order to apply Ortega's variant of Moser's twist theorem and Pei's theorem on the existence of Aubry-Mather sets. If compared with related works on regular potential, the presence of the singularity supposes an additional difficulty.