On quasi-periodic perturbations of hyperbolic-type degenerateequilibrium point of a class of planar systems

In this paper we consider two-dimensional nonlinear quasi-periodic system with small perturbations. Assume that the unperturbed system has a hyperbolic-type degenerate equilibrium point and the frequency satisfies the Diophantine conditions. Using the KAM iteration we prove that for sufficiently small perturbations, the system can be reduced by a nonlinear quasi-periodic transformation to a suitable normal form with an equilibrium point at the origin. Hence, for the system we can obtain a small quasi-periodic solution.