On the Persistence of Lower-Dimensional Tori in Reversible Systems with Hyperbolic-Type Degenerate Equilibrium Point Under Small Perturbations

Abstract

This paper focuses on the persistence of lower-dimensional tori in reversible systems with hyperbolic-type degenerate equilibrium point under small perturbations. Moreover, the dimension of degenerate variable is greater than or equal to 2. By KAM iteration and the Topological degree theorem, we prove that the invariant torus with given frequency persists under small perturbations.

Appendix

In this section we formulate a lemma which have been used in the previous section.

Let $\mathcal{U}_{s}$ denote the space of all real analytic functions $f(x)$ defined in the complex domain $D(s)= \{ x |\ |\text{Im } x|\leq s \}$; that is

$$ \mathcal{U}_{s}=\Big\{ f(x)\ \big| \ f(x )=\sum _{k\in \mathbb{Z}^{n} }f_{k} e^{\sqrt{-1}\langle k,x\rangle }, \|f\|_{s}< \infty \Big\} . $$

Let

$$ \mathcal{U}_{s}^{0}=\Big\{ f(x)\ \big| \ f(x ) \in \mathcal{K}_{s}, \ [f]=0 \Big\} . $$

Lemma 2

Suppose that $\omega$ satisfies the Diophantine condition $|\langle k, \omega \rangle |\geq \frac{\alpha }{|k|^{\tau }} , \ \forall k\in \mathbb{Z}^{n} \backslash \{0\}$. Then the equation

$$ \partial _{\omega } h(x) =g(x), \ g(x)\in \mathcal{U}_{s}^{0}, $$

has a unique solution $h(x)\in \bigcup _{0<\rho <s}\mathcal{U}_{s-\rho }^{0}$ with

$$ \|h\|_{s-\rho }\leq \frac{c}{\alpha \rho ^{\tau }} \|g\|_{s },\ 0< \rho < s, $$

where the constant $c$ depends only on $n$ and $\tau$.

For this lemma, we refer to Lemma 1 in [14].

Lemma 3

Let

$$ A=(a_{ij})_{p\times p}, \ B=(b_{ij})_{p\times p}, C=(c_{ij})_{p \times p}, X=(x_{ij})_{p\times p}. $$

Then the matrix equation $AX + XB = C$ is equivalent to the following linear equation $D\{X\} = \{C\}$, where

$$\begin{aligned} &\{X\}=( x_{11},x_{12},\ldots ,x_{1p},x_{21},x_{22},\ldots ,x_{2p}, \ldots ,x_{p1},x_{p2},\ldots ,x_{pp})^{T}, \\ &\{C\}=( c_{11},c_{12},\ldots ,c_{1p},c_{21},x_{22},\ldots ,x_{2p}, \ldots ,c_{p1},x_{p2},\ldots ,c_{pp})^{T} \end{aligned}$$

are $p^{2}$-columns and the coefficient matrix

$$ D =\displaystyle \left ( \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} a_{11}I_{p}+B^{T}& a_{12}I_{p} & \cdots & a_{1p}I_{p} \\ a_{21}I_{p}& a_{22}I_{p}+B^{T} & \cdots & a_{2p}I_{p} \\ \vdots & \vdots & \cdots & \vdots \\ a_{p1}I_{p}& a_{p2} & \cdots & a_{pp}I_{p}+B^{T} \\ \end{array}\displaystyle \right ) $$

with the eigenvalues

$$ \Big\{ \lambda _{i}^{a}+\lambda _{j}^{b}\big| i,j =1,2,\ldots ,p \Big\} , $$

if $A$ has the eigenvalues $\lambda _{1}^{a},\lambda _{2}^{a},\ldots ,\lambda _{p}^{a}$ and $B$ has the eigenvalues $\lambda _{1}^{b},\lambda _{2}^{b},\ldots ,\lambda _{p}^{b}$.

By direct verification, the conclusion of this lemma is obvious, and we will omit the proof of Lemma 3.