Persistence Of Invariant Tori In Generalized Hamiltonian Systems Under Weak Non-resonant Condition

In this paper,we adopt the method of rational approximation to prove the persistence of invariant tori under Brjuno-Russmann condition.The Brjuno-Riissmann non-resonant condition is|<k,ω>|≥α/△(|K|),0≠k∈Zn where α>0,Russmann approximation function △:[1,+∞)→[1,+∞)is a continuous increasing unbounded function and satisfies △(1)= 1,The method of rational approximation avoids the problem of small denominator,when solving the Homological function equation,and making the KAM iteration converge at the rate of qnε(0<q<1),rather than at the speed of the super-exponential function ελn(1<λ<2).