On Small Perturbation Of Two-dimensional Quasi-periodic Systems With An Elliptic-type Degenerate Equilibrium Point

In this thesis,we mainly study small perturbations of two-dimensional quasiperiodic systems with an elliptic-type degenerate equilibrium point.In some problems of physics,there are many quasi-periodic phenomena.These phenomena can be transformed into Hamiltonian systems.KAM theory is a very effective method to investigate quasi-periodic phenomena.In this thesis,the existence of quasi-periodic solutions for a dissipative planar system with an elliptic degenerate equilibrium point under quasi-periodic perturbations is studied by using the KAM theory.In order to discuss the existence of quasi-periodic solutions,we firstly transform the studied system into a normal form by some changes.Including a scaling change,then give a KAM iterative lemma,for the normal form,which is used to prove the existence of quasi-periodic solutions to the perturbed system.Because there are resonances between the tangent(internal)frequencies and normal(external)frequencies,we must dominate small denominators when solving homology equations on the KAM step,which means we need to estimate to the measure of the removed parameter set.