Persistence Of Invariant Tori Of Systems Under Weaker Nonresonance Conditions

Since the day KAM theory was established,it has played an imporant role in quantum mechanics,celestial mechanics and other subjects and gradually became one of the most remarkable mathematical achievement in 20 th century.This paper is devoted to studying the persistence of invariant tori of some systems under the weaker nonresonance conditions.Firstly,this paper studies the persistence of invariant tori with a given frequency for a system defined by a vector field on a torus.By introducing and adjusting the external parameter to eliminate the frequency drifts and using the polynomial structures of functions to truncate,it is proved that the torus with the given frequency persists under small perturbation if the frequency satisfies the Brjuno-Russmann(5)(5)nonresonance condition and the Brouwer topological degree of the frequency mapping is nonzero.Then,the paper uses the same mothod to study the Hamiltonian system with quasiperiodic perturbations and shows the persistence of invariant tori with given frequency provided that the given frequency satisfies Brjuno-Russmann(5)(5)nonresonance condition and the Brouwer topological degree of the frequency mapping is nonzero.At last,the paper continues to discuss the Hamiltonian system with quasiperiodic perturbations.In the past,the researches on this system are mainly focused on Russmann(5)(5)nondegenerate condition,but now this paper is committed to considering this system in classcial Kolmogorov nondegenerate case.By taking the frequency as an independent parameter directly and using the related properties of the approximation function,the paper proves that when the frequency mapping satisfies the weaker nonresonance condition,most of invariant tori of this system can persist under small peturbations.