Invariant surfaces of periodic systems with conservative cubic first approximation

Abstract

We study simultaneously two classes of time periodic systems of ODEs with small parameter " 0 : systems with “fast” and “slow” time. Their corresponding unperturbed systems x˙ i = − iyi" , y˙i = i(x3 i − ixi)" (i = 1, n, = 0, 1) have from one to 3n equilibrium points. We give explicit conditions on perturbations independent from parameter, which in both cases guarantee the existence of a certain number of invariant (n + 1)-dimensional surfaces, which are homeomorphic to tori for all sufficiently small values of the parameter. We provide explicit formulae of these surfaces, their asymptotic expansions and collections of systems with six invariant surfaces.