A Result on the Quasi-periodic Solutions of Forced Isochronous Oscillators at Resonance

In this paper, the authors are concerned with the forced isochronous oscillators with a repulsive singularity and a bounded nonlinearity x '' + V'(x) + g( x) = e(t, x, x'), where the assumptions on V, g and e are regular, described precisely in the introduction. Using a variant of Moser's twist theorem of invariant curves, the authors show the existence of quasi-periodic solutions and boundedness of all solutions. This extends the result of Liu to the case of the above system where e depends on the velocity.