A result on the quasi-periodic solutions of forced isochronous oscillators at resonance

Abstract

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.