A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems

Abstract

In this paper we consider the persistence of elliptic lower dimensional invariant tori with one normal frequency in reversible systems, andprove that if the frequency mapping $ω(y) ∈ \mathbb{R}^n$ and normal frequency mapping $λ(y) ∈ \mathbb{R}$ satisfy that $\text{deg} (ω/λ ,\mathcal{O},ω_0/λ_0)≠ 0,$ where $ω_0 =ω(y_0)$ and $λ_0 = λ(y_0)$ satisfy Melnikov's non-resonance conditions for some $y_0∈\mathcal{O}$, 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 integrablereversible systems, Discrete and Continuous Dynamical Systems series B, 14 (2010), 1237-1249].