Response Solutions of Degenerate Quasi-Periodic Systems Under Small Perturbations

Abstract

This paper considers a class of n-dimensional degenerate systems with a quasi-periodic perturbation whose frequency is a diophantine vector. Assume that the unperturbed system has the origin as an equilibrium, which is degenerate at one direction. By KAM iteration, we prove that the perturbed quasi-periodic system has a small response solution for sufficiently small perturbations. The proof is based on the idea of reducibility and the KAM technique of introducing parameters.

Appendix

1.1 Some Propeties of Functions

Lemma 4.1

(Lemma 5.1 [20]) Suppose $P(z,\theta )=\sum _{\vert \beta \vert \ge m}P_{\beta }(\theta )z^{\beta }$ is analytic on $D_{r,s}$. Let $\Vert P\Vert _{D_{r,s}}\le T$. If $\varepsilon \le \frac{r}{2}$, then $\Vert P\Vert _{D_{\varepsilon ,s}}\lessdot T(\frac{\varepsilon }{r})^{m}$.

1.2 Estimate for Implicit Function

Lemma 4.2

Consider

$$\begin{aligned} N(\mu )=l(\xi )-\lambda +{\tilde{N}}(\mu ),\ \mu =(\xi ,\lambda )\in M\subset \mathbb {C}^{2}, \end{aligned}$$

where $M=B(\Gamma _{-}, 3\varepsilon _{-})\cap (I_{\gamma _{-}/2+\varepsilon _{-}}\times \mathbb {C}).$ Let ${\tilde{N}}$ be real analytic and

$$\begin{aligned} \Vert \partial _{\mu }{\tilde{N}}\Vert _{M}=\Vert {\tilde{N}}_{\xi }\Vert _{M}+\Vert \tilde{N}_{\lambda }\Vert _{M}\le \frac{1}{2}. \end{aligned}$$

Moreover, $N(\xi ,\lambda )=0$ defines a real analytic implicit function on M

$$\begin{aligned} \Gamma :\lambda =\lambda (\xi ),\ \xi \in I_{\gamma _{-}/2} \end{aligned}$$

such that the neighborhood of $\Gamma$

$$\begin{aligned} B(\Gamma ,\varepsilon _{-})\subset M, \end{aligned}$$

where $\varepsilon _{-}, \gamma _{-}$ are the iteration parameters in the last step.

Consider

$$\begin{aligned} N_{+}(\mu )=l(\xi )-\lambda +{\tilde{N}}_{+}(\mu ), \end{aligned}$$

where ${\tilde{N}}_{+}={\tilde{N}}+{\hat{N}}$ with $\Vert {\hat{N}}\Vert _M\lessdot \varepsilon ^{2}.$

If $\varepsilon \le \varepsilon _{-}/4$ and $\varepsilon \le \gamma /2$, then there exists complex region $M_{+}\subset M\subset \mathbb {C}^{2}$ satisfing

$$\begin{aligned} dist(M_{+},\partial M)\ge \varepsilon _{-}/4, \end{aligned}$$

and $N_{+}$ is real analytic on $M_{+}$. Furthermore, ${\hat{N}}$ satisfies

$$\begin{aligned} \Vert \partial _{\mu }{\hat{N}}\Vert _{M_{+}}\lessdot \varepsilon ^{2}/\varepsilon _{-}. \end{aligned}$$

If $\Vert \partial _{\mu } {\tilde{N}}_{+}\Vert _{M_{+}}\le \frac{1}{2}$, then $N_{+}(\xi ,\lambda )=0$ determines a real analytic implicit function on $M_{+}$

$$\begin{aligned} \Gamma _{+}:\lambda =\lambda _{+}(\xi ),\ \xi \in I_{\gamma /2} \end{aligned}$$

such that

$$\begin{aligned} \vert \lambda _{+}(\xi )-\lambda (\xi )\vert \le 2\varepsilon ^{2},\ \xi \in I_{\gamma /2},\ B(\Gamma _{+},\varepsilon )\subset M_{+}. \end{aligned}$$

Moreover,

$$\begin{aligned} \vert N_{+}(\mu )\vert \le 8\varepsilon ,\ \mu \in M_+. \end{aligned}$$

Proof

Let $M_+=B(\Gamma ,3\varepsilon )\cap (I_{\gamma /2+\varepsilon }\times \mathbb {C})$. Since $\varepsilon \le \varepsilon _{-}/4$ and $\varepsilon \le \gamma /2$, by $B(\Gamma ,\varepsilon _{-})\subset M$ it follows easily that $M_{+}\subset M$ and dist$(M_{+}, \partial {M})\ge \varepsilon _{-}/4$. Using Cauchy’s estimate we have

$$\begin{aligned} \Vert \partial _{\mu }{\hat{N}}\Vert _{M_{+}}\lessdot \varepsilon ^{2}/\varepsilon _{-}. \end{aligned}$$

By $\Vert \partial _{\mu } {\tilde{N}}_{+}\Vert _{M_{+}}\le \frac{1}{2}$, the equation

$$\begin{aligned} N_{+}(\mu )=l(\xi )-\lambda +{\tilde{N}}_{+}(\mu )=0 \end{aligned}$$

determines implicitly an analytic curve on $M_{+}$

$$\begin{aligned} \Gamma _{+}:\lambda =\lambda _{+}(\xi ),\ \xi \in I_{\gamma /2}. \end{aligned}$$

By $\Vert \partial _{\mu }{\tilde{N}}\Vert _{M}\le \frac{1}{2}$, it follows that

$$\begin{aligned} \vert \lambda _{+}(\xi )-\lambda (\xi )\vert \le 2\varepsilon ^{2},\ \xi \in I_{\gamma /2}. \end{aligned}$$

Thus $B(\Gamma _{+},\varepsilon )\subset M_{+}.$ Let $\varepsilon <1/2$. For $(\xi , \lambda )\in M_{+}$, we have

$$\begin{aligned}\vert \lambda -\lambda _{+}(\xi )\vert \le \vert \lambda -\lambda (\xi )\vert +\vert \lambda (\xi )-\lambda _{+}(\xi )\vert \le 4\varepsilon . \end{aligned}$$

Noting that $\vert N_{+\lambda }\vert \le 2$, for all $\mu \in M_{+}$ and $N_{+}(\mu )=0$, for all $\mu =(\xi , \lambda (\xi ))$, we have

$$\begin{aligned} \vert N_{+}(\mu )\vert \le 8\varepsilon ,\ \mu \in M_+. \end{aligned}$$

Thus Lemma 4.2 is proved. $\square$

1.3 Implicit Function Theorem

The following implicit function theorem is given for solving the homological equation.

Lemma 4.3

Suppose that the eigenvalues of $A_{0}$ have non-zero real parts. Let $F(\mu ;\theta )\in C^{a}(M\times {\mathbb {T}}_{s})$. Then for sufficiently small $\varepsilon _{0}$ such that for $Q(\mu ;\theta )\in C^{a}(M\times {\mathbb {T}}_{s})$ and $\Vert Q\Vert _{M\times {\mathbb {T}}_{s}}\le c\varepsilon _{0}$, the equation

$$\begin{aligned} \partial _{\omega }W(\mu ;\theta )-A_{0}W(\mu ;\theta )-Q(\mu ;\theta )W(\mu ;\theta )=F(\mu ;\theta ) \end{aligned}$$

(4.1)

has a unique solution $W(\mu ;\theta )\in C^{a}(M\times {\mathbb {T}}_{s})$ and $\Vert W\Vert _{M\times {\mathbb {T}}_{s}}\le \tilde{c}\Vert F\Vert _{M\times {\mathbb {T}}_{s}}$, where $c, \tilde{c}$ are constant.

Proof

$W(\mu ;\theta )=(w_{1}(\mu ;\theta ), \cdots , w_{n}(\mu ;\theta ))^{T}\in C^{a}(M\times {\mathbb {T}}_{s})$. We define an operator $T: C^{a}(M\times {\mathbb {T}}_{s})\rightarrow C^{a}(M\times {\mathbb {T}}_{s})$ by

$$\begin{aligned} T(W(\mu ;\theta )):=\partial _{\omega }W(\mu ;\theta )-A_{0}W(\mu ;\theta )=\sum _{k\in {\mathbb {Z}}^{m}}({\textrm{i}}\langle k, \omega \rangle I_{n}-A_{0})W_{k}(\mu )e^{{\textrm{i}}\langle k, \theta \rangle }. \end{aligned}$$

Then the Eq. (4.1) can be written as

$$\begin{aligned} TW(\mu ;\theta )-Q(\mu ;\theta )W(\mu ;\theta )=F(\mu ;\theta ). \end{aligned}$$

For simplicity, let $A_{0}$ possess a Jordan form

$$\begin{aligned} J_{A_{0}}=\text{ diag }(A_{01},\cdots ,A_{0l}), \end{aligned}$$

where $A_{0i}$ is at most two-order because the case of higher order is similar. Then $A_{0i}=\text{ diag }(\zeta _{i},\bar{\zeta _{i}})$, or $\zeta _{i}$, or $\zeta _{i}I_{2}$, or $\zeta _{i}I_{2}+J_{0}$ for $1\le i\le l$, where $\vert {\textrm{Re}}(\zeta _{i})\vert \ge b_{0}>0$ and $J_{0}=\begin{pmatrix} 0 &{}\quad 0 \\ 1 &{} 0 \end{pmatrix}$. By the definition of T, the eigenvalues corresponding to T are ${{\textrm{i}}}\langle k, \omega \rangle +\zeta _{i}$. Noting $\vert \text {Re}(\zeta _{i})\vert \ge b_{0}$, then we can get T is invertible and $T^{-1}:C^{a}(M\times {\mathbb {T}}_{s})\rightarrow C^{a}(M\times {\mathbb {T}}_{s})$

$$\begin{aligned} T^{-1}(W(\mu ;\theta ))=\sum _{k\in {\mathbb {Z}}^{m}}({\textrm{i}}\langle k, \omega \rangle I_{n}-A_{0})^{-1}W_{k}(\mu )e^{{\textrm{i}}\langle k, \theta \rangle }. \end{aligned}$$

One can check that

$$\begin{aligned} \Vert T^{-1}\Vert =\sup _{0\ne W(\mu ;\theta )\in C^{a}(M\times {\mathbb {T}}_{s})}\frac{\Vert T^{-1}(W(\mu ;\theta ))\Vert }{\Vert W(\mu ;\theta )\Vert }\le \sup _{0\ne W(\mu ;\theta )\in C^{a}(M\times {\mathbb {T}}_{s})}\frac{c_{1}\Vert W(\mu ;\theta )\Vert }{\Vert W(\mu ;\theta )\Vert }=c_{1}, \end{aligned}$$

where $c_{1}=n\times \max \{\frac{1}{b_{0}},\frac{1}{b_{0}^{2}}\}$.

Define $L: C^{a}(M\times {\mathbb {T}}_{s})\rightarrow C^{a}(M\times {\mathbb {T}}_{s})$ by

$$\begin{aligned} L(W(\mu ;\theta )):=T^{-1}(F(\mu ;\theta )+Q(\mu ;\theta )W(\mu ;\theta )). \end{aligned}$$

Below we prove that L has a fixed point. For any $W_{1}(\mu ;\theta ), W_{2}(\mu ;\theta )\in C^{a}(M\times {\mathbb {T}}_{s})$,

$$\begin{aligned} \Vert LW_{1}-LW_{2}\Vert{} & {} =\Vert T^{-1}Q(W_{1}-W_{2})\Vert \le \Vert T^{-1}\Vert \Vert Q\Vert \Vert W_{1}-W_{2}\Vert \\{} & {} \le c_{1}c\varepsilon _{0}\Vert W_{1}-W_{2}\Vert \le \frac{1}{2}\Vert W_{1}-W_{2}\Vert . \end{aligned}$$

By Banach fixed point theorem, L has a unique fixed point $W_{*}$ in $C^{a}(M\times {\mathbb {T}}_{s})$. Moreover,

$$\begin{aligned} \Vert W_{*}(\mu ;\theta )\Vert{} & {} =\Vert T^{-1}(F(\mu ;\theta )+Q(\mu ;\theta )W_{*}(\mu ;\theta ))\Vert \le \Vert T^{-1}\Vert \Vert F(\mu ;\theta )\\{} & {} \quad +Q(\mu ;\theta )W_{*}(\mu ;\theta )\Vert \le \Vert T^{-1}\Vert (\Vert F(\mu ;\theta )\Vert \\{} & {} \quad +\Vert Q(\mu ;\theta )\Vert \Vert W_{*}(\mu ;\theta )\Vert )\le c_{1}(\Vert F(\mu ;\theta )\Vert +c\varepsilon _{0}\Vert W_{*}(\mu ;\theta )\Vert ), \end{aligned}$$

thus we get that

$$\begin{aligned} \Vert W_{*}(\mu ;\theta )\Vert \le \frac{c_{1}}{1-c_{1}c\varepsilon _{0}}\Vert F(\mu ;\theta )\Vert \le \tilde{c}\Vert F(\mu ;\theta )\Vert , \end{aligned}$$

where $\tilde{c}=\frac{c_{1}}{1-c_{1}c\varepsilon _{0}}$. Thus Lemma 4.3 is proved. $\square$