Boundedness of solutions for polynomial potentials withC ² time dependent coefficients

Abstract

In this paper we prove the existence of invariant curves and thus stability for all time for a class of Hamiltonian systems with time dependent potentials:

$$\frac{{d^2 x}}{{dt^2 }} + Vx(x,t) = 0,x \in R^1 $$

where$\begin{gathered} V(x,t) = \tfrac{1}{{2n + 2}}x^{2n + 2} + \Sigma _{j = 0}^{2n} \tfrac{{Pj(t)}}{{j + 1}}x^{j + 1} ,p_j (t + 1) = p_j (t),p_j \in C^2 ,2n \geqslant j \geqslant n + 1;p_j \in \hfill \\ C^1 ,n \geqslant j \geqslant 0,n \geqslant 1. \hfill \\ \end{gathered}$