Elliptic-type degenerate invariant tori for quasi-periodically forced four-dimensional non-conservative systems

Abstract

In this paper, we will focus on the persistence of elliptic-type degenerate invariant tori with the same frequency vector ω∈Rd as the forcing in a class of quasi-periodically forced four-dimensional non-conservative systems. It is shown that, under suitable hypothesis of smoothness with respect to ϵ and the Brjuno–Rüssmann's non-resonant condition instead of ordinary Diophantine condition with respect to the frequency vector ω∈Rd, the invariant tori persist under small perturbations for most of the sufficiently small parameters ϵ in the sense of the Lebesgue measure. Hence, the system has a quasi-periodic solution with the same frequency vector ω as the forcing. The proof is based on the Pöschel–Rüssmann KAM method, which is a KAM-type method in which one uses the polynomial structure of function to truncate, introduce a parameter q and make the steps of KAM iteration infinitely small in the speed of function qnκ,0<q<1, rather than super exponential function like traditional KAM method.