Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies

Zhang, Dongfeng; Xu, Ji‐Qing; Xu, Xindong

doi:10.3934/dcds.2018123

Discrete &Amp; Continuous Dynamical Systems - A

2018

DOI: 10.3934/dcds.2018123

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Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies

Dongfeng Zhang¹,

Ji‐Qing Xu²,

Xindong Xu³

Abstract: 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) … Show more

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Cited by 9 publications

(2 citation statements)

References 34 publications

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“…For high dimensional quasi-periodic systems, Zhou-Wang [51] used periodic approximation to study the reducibility of quasi-periodic GL(d, R) cocycles with Liouvillean basic frequencies. By use of the special structure of skew symmetric systems, Zhang-Xu-Xu [48] obtained the reducibility of three dimensional skew symmetric systems with Liouvillean basic frequencies. Comparing to [51], the former is discrete, our system is continuous, so the method of periodic approximation for discrete case cannot be adopted to continuous case.…”

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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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confidence: 99%

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

Zhang¹,

Xu²

2022

CPAA

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“…For high-dimensional quasi-periodic systems, Zhou and Wang [40] used periodic approximation to study the reducibility of quasi-periodic GL(d, R) cocycles with Liouvillean frequencies. Zhang, Xu and Xu [38] obtained the reducibility of a threedimensional skew-symmetric linear system with Liouvillean basic frequencies. Compared to [40], the former is discrete and linear, while our system is continuous and nonlinear, so there are essential obstructions in applying the method of periodic approximation for the discrete case in [40] to the continuous case.…”

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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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Reducibility of Three Dimensional Skew Symmetric System with High Dimensional Weak Liouvillean Frequencies

Liu,

Shan,

Wang

2024

Acta. Math. Sin.-English Ser.

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Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies

Cited by 9 publications

References 34 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 a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies

Reducibility of Three Dimensional Skew Symmetric System with High Dimensional Weak Liouvillean Frequencies

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