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.
We are concerned with the existence of quasi-periodic solutions for the following equation x" + Fx (x, t)x' + ω2x + φ(x,t) = 0,where F and φ are smooth functions and 2π-periodic in t, ω＞ 0 is a constant. Under some assumptions on the parities of F and φ, we show that the Dancer's function, which is used to study the existence of periodic solutions, also plays a role for the existence of quasi-periodic solutions and the Lagrangian stability (i.e. all solutions are bounded).
We deal with the stability of zero solutions of planar Hamiltonian and reversible systems which are quasi-periodic in the time variable. Under some reasonable assumptionswe prove the existence of quasi-periodic solutions in a small neighborhood of zero solutions and the stability of zero solutions.
LIU Bin Laboratory of Mathematics and its Applications School of Mathematical Sciences Peking University Beijing 100871China
