A KAM THEOREM FOR THE ELLIPTIC LOWER DIMENSIONAL TORT WITH ONE NORMAL FREQUENCY IN REVERSIBLE SYSTEMS

In this paper we consider the persistence of elliptic lower dimensional invariant tori with one normal frequency in reversible systems, and prove that if the frequency mapping w(y) is an element of R-n and normal frequency mapping lambda(y) is an element of R satisfy that deg(omega/lambda, O, omega(0)/lambda(0)) not equal 0, where omega(0) = omega(y(0)) and lambda(0) = lambda(y(0)) satisfy Melnikov's non -resonance conditions for some y(0) is an element of O, then the direction of this frequency for the invariant torus persists under small perturbations. Our result is a generalization of X. Wang et al[Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems, Discrete and Continuous Dynamical Systems series B, 14 (2010), 1237-1249].