On invariant tori in some reversible systems

Abstract

In the present paper, we consider the following reversible system \begin{equation*} \begin{cases} \dot{x}=ω_0+f(x,y),\\ \dot{y}=g(x,y), \end{cases} \end{equation*} where $x\in {\mathbf{T}}^d$, $y\backsim 0\in {\mathbf{R}}^d$, ${\omega }_0$ is Diophantine, $f(x,y)=O(y)$, $g(x,y)=O(y^2)$ and $f$, $g$ are reversible with respect to the involution G: $(x,y)\mapsto (-x,y)$, that is, $f(-x,y)=f(x,y)$, $g(-x,y)=-g(x,y)$. We study the accumulation of an analytic invariant torus ${\Gamma }_0$ of the reversible system with Diophantine frequency ${\omega }_0$ by other invariant tori. We will prove that if the Birkhoff normal form around ${\Gamma }_0$ is 0-degenerate, then ${\Gamma }_0$ 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 ${\Gamma }_0$ being one. We will also prove that if the Birkhoff normal form around ${\Gamma }_0$ is $j$-degenerate ($1\le j\le d-1$) and condition (1.6) is satisfied, then through ${\Gamma }_0$ there passes an analytic subvariety of dimension $d+j$ foliated into analytic invariant tori with frequency vector ${\omega }_0$. If the Birkhoff normal form around ${\Gamma }_0$ is $d-1$-degenerate, we will prove a stronger result, that is, a full neighborhood of ${\Gamma }_0$ is foliated into analytic invariant tori with frequency vectors proportional to ${\omega }_0$.