On invariant tori in some reversible systems

In the present paper, we consider the reversible system ẋ=ω0+f(x,y), ẏ=g(x,y), where x ∈ Td, y ∽ 0 ∈ Rd, ω0 is Diophantine, f(x, y) = O(y), g(x, y) = O(y2) 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 Γ0 of the reversible system with Diophantine frequency ω0 by other invariant tori. We will prove that if the Birkhoff normal form around Γ0 is 0-degenerate, then Γ0 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 Γ0 being one. We will also prove that if the Birkhoff normal form around Γ0 is j-degenerate (1 ≤ j ≤ d − 1) and condition (1.6) is satisfied, then through Γ0 there passes an analytic subvariety of dimension d + j foliated into analytic invariant tori with frequency vector ω0. If the Birkhoff normal form around Γ0 is d − 1-degenerate, we will prove a stronger result, that is, a full neighborhood of Γ0 is foliated into analytic invariant tori with frequency vectors proportional to ω0.