Construction of response solutions for two classes of quasi-periodically forced four-dimensional nonlinear systems with degenerate equilibrium point under small perturbations

By developing two KAM theorems, in this paper, we show that two classes of quasi-periodically forced four-dimensional nonlinear systems with degenerate equilibrium point, a reversible system and a non-conservative system, admit a response solution under small perturbations. For the degenerate reversible system, applying special structure of unperturbed nonlinear term and Herman method, we successfully control the shift of equilibrium point, which is difficult in view of the degenerate linear term. For the degenerate non-conservative system, KAM method is brought into force even in completely degenerate case because of the restrictions on the smallness and average of perturbation. Moreover, arithmetic condition on the frequency is assumed to satisfy the Brjuno-Rüssmann's non-resonant condition. By the Pöschel-Rüssmann KAM method, we prove that these two kinds of perturbed systems can be reduced to a suitable normal form with zero as equilibrium point by a quasi-periodic transformation.