A New KAM Theorem for the Hyperbolic Lower Dimensional Tori in Reversible Systems

Abstract

This paper puts forward a new KAM theorem for the hyperbolic lower dimensional tori in reversible system without assuming any non-degeneracy condition, which can be viewed as a unified form under all kinds of non-degenerate conditions, including Kolmogorov, Bruno and Rüssmann non-degeneracy condition. The theorem with some non-degeneracy conditions not only includes many previous achievements on KAM theory in reversible systems, but also gives some interesting results on the persistence of hyperbolic lower dimensional invariant tori with prescribed frequency in reversible system.

Appendix

In this section we formulate a lemma which have been used in the previous section. Let $\mathcal{B}_{s}$ denote the space of all real analytic functions $f(x)$ defined in the complex domain $D(s)=\{x:|\operatorname{Im} x|\leq s\}$; that is

$$\mathcal{B}_{s}= \bigl\{ f(x,\xi) \bigm| f(x,\xi)=\sum_{k}f_{k}(\xi) e^{\sqrt {-1}\langle k,x\rangle},\ \|f\|_{\mathcal{O}\times D(s,r)} ^{\alpha }< \infty\bigr\} . $$

Then it is easy to see that $\mathcal{B}_{s}$ is a Banach space.

Lemma 2

Let $\lambda_{1}(\xi),\ldots,\lambda_{2p}(\xi)$ be eigenvalues of the matrix $\varOmega(\xi)$ such that

$$ \bigl|\operatorname{Re}\lambda_{i} (\xi)\bigr|\geq\sigma,\quad \textit{for } \forall\xi\in\mathcal{O},\ i=1,2,\ldots,2p. $$

(3.1)

Then, there exists a sufficiently small $\epsilon_{0}$ depending only on $\varOmega$ such that for any $\tilde{\varOmega}(x)\in\mathcal{B}_{s}$, $g(x)\in\mathcal{B}_{s}$, if $\|\tilde{\varOmega} \|^{\alpha}_{\mathcal{O}\times D(s,r) }\leq \epsilon_{0}$, the equation

$$\partial_{\omega}f(x) - (\varOmega +\tilde{\varOmega}(x))f(x)=g(x) $$

has a solution $f(x)\in\mathcal{B}_{s}$ with

$$\|f\|^{\alpha}_{\mathcal{O}\times D(s,r) }\leq c \|g\|^{\alpha}_{\mathcal {O}\times D(s,r) }, $$

where the constant $c$ depends only on $\varOmega$.

For this lemma, we refer to Lemma 3.2 in [24].