On invariant tori in some reversible systems

Abstract

In the present paper, we consider the following reversible system ẋ = ω 0 + f (x, y), ẏ = g(x, y),) and f , g are reversible with respect to the involution G: (x, y) → (−x, y), that is, f (−x, y) = f (x, y), g(−x, y) = −g(x, y). We study the accumulation of an analytic invariant torus Γ 0 of the reversible system with Diophantine frequency ω 0 by other invariant tori. We will prove that if the Birkhoff normal form around Γ 0 is 0-degenerate, then Γ 0 is accumulated by other analytic invariant tori, the Lebesgue measure of the union of these tori being positive and the density of the union of these tori at Γ 0 being one. We will also prove that if the Birkhoff normal form around Γ 0 is j-degenerate (1 ≤ j ≤ d − 1) and condition (1.6) is satisfied, then through Γ 0 there passes an analytic subvariety of dimension d + j foliated into analytic invariant tori with frequency vector ω 0 . If the Birkhoff normal form around Γ 0 is d − 1-degenerate, we will prove a stronger result, that is, a full neighborhood of Γ 0 is foliated into analytic invariant tori with frequency vectors proportional to ω 0 .