Persistence Of Invariant Tori Of Reversible Systems

In the past, KAM theorems of reversible systems generally require re-versible systems to meet some non-degeneracy condition and diophantine condition. In this paper, we weaken non-degeneracy condition and diophantine condition to ob-tain the persistence of invariant tori of reversible systems. First, we use the special nature that the dimension of the frequency w is 2 and the modified KAM iteration to prove the persistence of lower dimensional hyperbolic invariant tori of reversible systems without non-degeneracy condition, but the frequency has some small shifts from small perturbations. Moreover, we prove the persistence of invariant tori for a class of reversible systems under the Brjuno-Russmann non-resonance condition. The Brjuno-Russmann non-resonance condition is weaker than diophantine condi-tion. In the proof, we use the polynomial structure of function to truncate and introduce a parameter q to make the steps of KAM iteration infinitely small in the speed of function qn∈,0<q<1, rather than super exponential function.