On invariant tori in some reversible systems

TL;DR

This paper investigates the structure and accumulation of invariant tori in reversible dynamical systems with Diophantine frequencies, establishing conditions under which these tori densely fill neighborhoods or form foliations.

Contribution

It provides new results on the accumulation and foliation of invariant tori in reversible systems based on the degeneracy of the Birkhoff normal form.

Findings

Invariant tori accumulate around certain tori when the Birkhoff normal form is 0-degenerate.

A positive measure of invariant tori densely fill neighborhoods of the original torus.

Foliations into invariant tori occur under specific degeneracy conditions of the Birkhoff normal form.

Abstract

In the present paper, we consider the following reversible system \begin{equation*} \begin{cases} \dot{x}=\omega_0+f(x,y),\\ \dot{y}=g(x,y), \end{cases} \end{equation*} where $x\in\mathbf{T}^{d}$, $y\backsim0\in \mathbf{R}^{d}$, $\omega_0$ 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 ($1\leq j\leq d-1$) 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 $\omega_0$.