Analytic Reducibility of Resonant Cocycles to a Normal Form

Chavaudret, Claire; Stolovitch, Laurent

doi:10.1017/s1474748014000383

Cited by 6 publications

(3 citation statements)

References 12 publications

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“…where [•] denotes its integer part, 0 < c < 1 is a constant which will be defined in (4.21), and c 0 = c 4 5 •10 τ is a constant depending on τ, c. We summarize one step of KAM iteration in the following Lemma. The key point is to guarantee the non-resonant condition in KAM iteration by adjusting some parameters [10,19,20,27,36,39].…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

On the reducibility of analytic quasi-periodic systems with Liouvillean basic frequencies

Zhang¹,

Xu²

2022

CPAA

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In this paper we consider the linear quasi-periodic system<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \dot{x} = (A+\epsilon P(t)) x, x\in \mathbb{R}^{d}, \end{equation*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ A $\end{document}</tex-math></inline-formula> is a <inline-formula><tex-math id="M2">\begin{document}$ d\times d $\end{document}</tex-math></inline-formula> constant matrix with elliptic type, <inline-formula><tex-math id="M3">\begin{document}$ P(t) $\end{document}</tex-math></inline-formula> is analytic quasi-periodic with respect to <inline-formula><tex-math id="M4">\begin{document}$ t $\end{document}</tex-math></inline-formula> with basic frequencies <inline-formula><tex-math id="M5">\begin{document}$ \omega = (1, \alpha), $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M6">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> being irrational, and <inline-formula><tex-math id="M7">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is a small perturbation parameter. If some suitable non-resonant conditions and non-degeneracy conditions hold, and the basic frequencies satisfy that <inline-formula><tex-math id="M8">\begin{document}$ 0\leq \beta(\alpha) < r, $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M9">\begin{document}$ \beta(\alpha) = \limsup\limits_{n\rightarrow \infty}\frac{\ln q_{n+1}}{q_{n}}, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M10">\begin{document}$ q_{n} $\end{document}</tex-math></inline-formula> is the sequence of denominations of the best rational approximations for <inline-formula><tex-math id="M11">\begin{document}$ \alpha \in \mathbb{R} \setminus\mathbb{Q}, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M12">\begin{document}$ r $\end{document}</tex-math></inline-formula> is the initial radius of analytic domain, it is proved that for most sufficiently small <inline-formula><tex-math id="M13">\begin{document}$ \epsilon, $\end{document}</tex-math></inline-formula> this system can be reduced to a constant system <inline-formula><tex-math id="M14">\begin{document}$ \dot{x} = A^{*}x, x\in \mathbb{R}^{d}, $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M15">\begin{document}$ A^{*} $\end{document}</tex-math></inline-formula> is a constant matrix close to <inline-formula><tex-math id="M16">\begin{document}$ A. $\end{document}</tex-math></inline-formula> As some applications, we apply our results to quasi-periodic Schrödinger equations with an external parameter to study the Lyapunov stability of the equilibrium and the existence of quasi-periodic solutions.

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

On the reducibility of analytic quasi-periodic systems with Liouvillean basic frequencies

Zhang¹,

Xu²

2022

CPAA

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“…Her-You [13] and Chavaudret [5] established the full measure reducibility with coefficients in other groups. For the latest reducibility results of resonant cocycles and infinite dimensional quasi-periodic systems, we refer to [2,3,6,11] and the references therein.…”

Section: Dongfeng Zhang Junxiang Xu and Xindong Xumentioning

confidence: 99%

Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies

Zhang¹,

Xu²,

Xu³

2018

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper we consider the systemẋ = (A() + m P (t;))x, x ∈ R 3 , where is a small parameter, A, P are all 3 × 3 skew symmetric matrices, A is a constant matrix with eigenvalues ±iλ() and 0, whereλ() = λ + am 0 m 0 + O(m 0 +1)(m 0 < m), am 0 = 0, P is a quasi-periodic matrix with basic frequencies ω = (1, α) with α being irrational. First, it is proved that for most of sufficiently small parameters, this system can be reduced to a rotation system. Furthermore, if the basic frequencies satisfy that 0 ≤ β(α) < r, where β(α) measures how Liouvillean α is, r is the initial analytic radius, it is proved that for most of sufficiently small parameters, this system can be reduced to constant system by means of a quasi-periodic change of variables.

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“…Zhao [39] proved the reducibility of nonlinear quasi-periodic system (1), when the eigenvalues of A are allowed to be multiple. Chavaudret [6] studied the reducibility of resonant cocycles. Moreover, the Diophantine condition (2) can also be weakened.…”

mentioning

confidence: 99%

Reducibility of a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies

Zhang¹,

Xu²

2020

Ergod. Th. Dynam. Sys.

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In this paper we consider the following nonlinear quasi-periodic system: $$\begin{eqnarray}{\dot{x}}=(A+\unicode[STIX]{x1D716}P(t,\unicode[STIX]{x1D716}))x+\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})+h(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $A$ is a $d\times d$ constant matrix of elliptic type, $\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})$ is a small perturbation with $\unicode[STIX]{x1D716}$ as a small parameter, $h(x,t,\unicode[STIX]{x1D716})=O(x^{2})$ as $x\rightarrow 0$ , and $P,g$ and $h$ are all analytic quasi-periodic in $t$ with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$ , where $\unicode[STIX]{x1D6FC}$ is irrational. It is proved that for most sufficiently small $\unicode[STIX]{x1D716}$ , the system is reducible to the following form: $$\begin{eqnarray}{\dot{x}}=(A+B_{\ast }(t))x+h_{\ast }(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $h_{\ast }(x,t,\unicode[STIX]{x1D716})=O(x^{2})~(x\rightarrow 0)$ is a high-order term. Therefore, the system has a quasi-periodic solution with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$ , such that it goes to zero when $\unicode[STIX]{x1D716}$ does.

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Analytic Reducibility of Resonant Cocycles to a Normal Form

Cited by 6 publications

References 12 publications

On the reducibility of analytic quasi-periodic systems with Liouvillean basic frequencies

On the reducibility of analytic quasi-periodic systems with Liouvillean basic frequencies

Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies

Reducibility of a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies

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