April  2022, 21(4): 1417-1445. doi: 10.3934/cpaa.2022024

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

School of Mathematics, Southeast University, Nanjing 210096, China

* Corresponding author

Received  April 2021 Revised  September 2021 Published  April 2022 Early access  February 2022

Fund Project: This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11001048, 11571072, 11771077, 11871146) and the Natural Science Foundation of Jiangsu Province, China (No. BK20201262)

In this paper we consider the linear quasi-periodic system
$ \begin{equation*} \dot{x} = (A+\epsilon P(t)) x, x\in \mathbb{R}^{d}, \end{equation*} $
where
$ A $
is a
$ d\times d $
constant matrix with elliptic type,
$ P(t) $
is analytic quasi-periodic with respect to
$ t $
with basic frequencies
$ \omega = (1, \alpha), $
with
$ \alpha $
being irrational, and
$ \epsilon $
is a small perturbation parameter. If some suitable non-resonant conditions and non-degeneracy conditions hold, and the basic frequencies satisfy that
$ 0\leq \beta(\alpha) < r, $
where
$ \beta(\alpha) = \limsup\limits_{n\rightarrow \infty}\frac{\ln q_{n+1}}{q_{n}}, $
$ q_{n} $
is the sequence of denominations of the best rational approximations for
$ \alpha \in \mathbb{R} \setminus\mathbb{Q}, $
$ r $
is the initial radius of analytic domain, it is proved that for most sufficiently small
$ \epsilon, $
this system can be reduced to a constant system
$ \dot{x} = A^{*}x, x\in \mathbb{R}^{d}, $
where
$ A^{*} $
is a constant matrix close to
$ A. $
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.
Citation: Dongfeng Zhang, Junxiang Xu. On the reducibility of analytic quasi-periodic systems with Liouvillean basic frequencies. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1417-1445. doi: 10.3934/cpaa.2022024
References:
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[2]

D. BambusiB. GrébertA. Maspero and D. Robert, Reducibility of the quantum harmonic oscillator in d-dimensions with polynomial time-dependent perturbation, Anal. Partial Differ. Equ., 11 (2018), 775-799.  doi: 10.2140/apde.2018.11.775.

[3]

D. Bambusi, Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations, I, Trans. Amer. Math. Soc., 370 (2018), 1823-1865.  doi: 10.1090/tran/7135.

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D. Bambusi, Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations, II, Comm. Math. Phys., 353 (2017), 353-378.  doi: 10.1007/s00220-016-2825-2.

[5]

Yu. N. Bibikov, On the stability of the zero solution of essentially nonlinear Hamiltonian systems and reversible systems with one degree of freedom, Differ. Equ., 38 (2002), 609-614.  doi: 10.1023/A:1020298221798.

[6]

A. Bounemoura, Effective stability for Gevrey and finitely differentiable prevalent Hamiltonians, Commun. Math. Phys., 307 (2011), 157-183.  doi: 10.1007/s00220-011-1306-x.

[7]

A. Bounemoura and J. Féjoz, KAM, $\alpha$-Gevrey regularity and the $\alpha$-Bruno-Rüssmann condition, Ann. Sc. Norm. Super. Pisa Cl. Sci., 19 (2019), 1225-1279. 

[8]

C. Chavaudret and S. Marmi, Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition, J. Mod. Dyn., 6 (2012), 59-78.  doi: 10.3934/jmd.2012.6.59.

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C. Chavaudret, Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles, Bull. Soc. Math. France, 141 (2013), 47-106.  doi: 10.24033/bsmf.2643.

[10]

C. Chavaudret and L. Stolovitch, Analytic reducibility of resonant cocycles to a normal form, J. Inst. Math. Jussieu, 15 (2016), 203-223.  doi: 10.1017/S1474748014000383.

[11]

E. I. Dinaburg and Ya. G. Sinai, The one dimensional Schrödinger equation with quasi-perioidc potential, Funct. Anal. Appl., 9 (1975), 8-21. 

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L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Commun. Math. Phys., 146 (1992), 447-482. 

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L. H. Eliasson, Almost reducibility of linear quasi-periodic systems, Smooth ergodic theory and its applications (Seattle, WA, 1999), 679–705, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001. doi: 10.1090/pspum/069/1858550.

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L. H. Eliasson and S. B. Kuksin, On reducibility of Schrödinger equations with quasi-periodic in time potentials, Commun. Math. Phys., 286 (2009), 125-135.  doi: 10.1007/s00220-008-0683-2.

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B. Fayad and R. Krikorian, Herman's last geometric theorem, Ann. Sci. Éc. Norm. Supér., 42 (2009), 193–219. doi: 10.24033/asens.2093.

[16]

H. Her and J. You, Full measure reducibility for generic one-parameter family of quasi-periodic linear systems, J. Dynam. Differ. Equ., 20 (2008), 831-866.  doi: 10.1007/s10884-008-9113-6.

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X. Hou and J. You, Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems, Invent. Math., 190 (2012), 209-260.  doi: 10.1007/s00222-012-0379-2.

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R. A. Johnson and G. R. Sell, Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems, J. Differ. Equ., 41 (1981), 262-288.  doi: 10.1016/0022-0396(81)90062-0.

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À. Jorba and C. Simó, On the reducibility of linear differential equation with quasi-perioidc coefficients, J. Differ. Equ., 98 (1992), 111-124.  doi: 10.1016/0022-0396(92)90107-X.

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À. Jorba and C. Simó, On quasi-periodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal., 27 (1996), 1704-1737.  doi: 10.1137/S0036141094276913.

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R. Krikorian, Global density of reducible quasi-periodic cocycles on $\mathbb{T}^{1} \times SU(2)$, Ann. Math., 154 (2001), 269-326.  doi: 10.2307/3062098.

[22]

B. Liu, The stability of the equilibrium of planar Hamiltonian and reversible systems, J. Dynam. Differ. Equ., 18 (2006), 975-990.  doi: 10.1007/s10884-006-9027-0.

[23]

B. Liu, The stability of equilibrium of quasi-periodic planar Hamiltonian and reversible systems, Sci. China Math., 53 (2010), 125-136.  doi: 10.1007/s11425-009-0117-4.

[24]

J. Lopes Dias, A normal form theorem for Brjuno skew systems through renormalization, J. Differ. Equ., 230 (2006), 1-23.  doi: 10.1016/j.jde.2006.07.021.

[25]

J. Lopes Dias, Brjuno condition and renormalization for Poincaré flows, Discrete Contin. Dyn. Syst., 15 (2006), 641-656.  doi: 10.3934/dcds.2006.15.641.

[26]

J. Lopes Dias and J. Pedro Gaivão, Linearization of Gevrey flows on $\mathbb{T}^{d}$ with a Brjuno type arithmetical condition, J. Differ. Equ., 267 (2019), 7167-7212.  doi: 10.1016/j.jde.2019.07.020.

[27]

J. Liang and J. Xu, A note on the extension of Dinaburg-Sinai theorem to higher dimension, Ergodic Theory Dynam. Syst., 25 (2005), 1539-1549.  doi: 10.1017/S0143385705000118.

[28]

W. Magnus and S. Winkler, Hill's Equation, Corrected reprint of the 1966 edition. Dover Publications, Inc., New York, 1979.

[29]

J. P. Marco and D. Sauzin, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems, Publ. Math. Inst. Hautes Études Sci., No. 96 (2002), 199–275 (2003). doi: 10.1007/s10240-003-0011-5.

[30]

J. Pöschel, KAM $\grave{a}$ la R, Regul. Chaotic Dyn., 16 (2011), 17-23.  doi: 10.1134/S1560354710520060.

[31]

G. Popov, KAM theorem for Gevrey Hamiltonians, Ergodic Theory Dynam. Systems, 24 (2004), 1753-1786.  doi: 10.1017/S0143385704000458.

[32]

H. Rüssmann, On the one dimensional Schrödinger equation with a quasi-perioidc potential, Ann. New York Acad. Sci., 357 (1980), 90-107. 

[33]

H. Rüssmann, Convergent transformations into a normal form in analytic Hamiltonian systems with two degrees of freedom on the zero energy surface near degenerate elliptic singularities, Ergodic Theory Dynam. Systems, 24 (2004), 1787-1832.  doi: 10.1017/S0143385703000774.

[34]

J. WangJ. You and Q. Zhou, Response solutions for quasi-periodically forced harmonic oscillators, Trans. Amer. Math. Soc., 369 (2017), 4251-4274.  doi: 10.1090/tran/6800.

[35]

X. Wang and J. Xu, On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point, Nonlinear Anal., 69 (2008), 2318-2329.  doi: 10.1016/j.na.2007.08.016.

[36]

X. WangJ. Xu and D. Zhang, On the persistence of degenerate lower-dimensional tori in reversible systems, Ergodic Theory Dynam. Systems, 35 (2015), 2311-2333.  doi: 10.1017/etds.2014.34.

[37]

Y. Wu and Y. Wang, The stability of the elliptic equilibuium of planar quasi-periodic Hamiltonian system, Acta Math. Sin., 28 (2002), 801-816.  doi: 10.1007/s10114-011-0006-y.

[38]

Zh. Wang and Zh. Liang, Reducibility of 1D quantum harmonic oscillator perturbed by a quasiperiodic potential with logarithmic decay, Nonlinearity, 30 (2017), 1405-1448.  doi: 10.1088/1361-6544/aa5d6c.

[39]

J. Xu and Q. Zheng, On the reducibility of linear differential equations with quasiperiodic coefficients which are degenerate, Proc. Amer. Math. Soc., 126 (1998), 1445-1451.  doi: 10.1090/S0002-9939-98-04523-7.

[40]

J. Xu, On the reducibility of a class of linear differential equation with quasi-periodic coefficients, Mathematika, 46 (1999), 443-451.  doi: 10.1112/S0025579300007907.

[41]

J. Xu and J. You, Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition, J. Math. Pures Appl., 80 (2001), 1045-1067.  doi: 10.1016/S0021-7824(01)01221-1.

[42]

J. Xu, Normal form of reversible systems and persistence of lower dimensional tori under weaker nonresonance conditions, SIAM J. Math. Anal., 36 (2004), 233-255.  doi: 10.1137/S0036141003421923.

[43]

J. Xu and X. Lu, On the reducibility of two-dimensional linear quasi-periodic systems with small parameter, Ergodic Theory Dynam. Systems, 35 (2015), 2334-2352.  doi: 10.1017/etds.2014.31.

[44]

X. Yuan and A. Nunes, A note on the reducibility of linear differential equations with quasiperiodic coefficients, Int. J. Math. Math. Sci., 2003, 4071–4083. doi: 10.1155/S0161171203206025.

[45]

D. Zhang and J. Xu, Invariant curves of analytic reversible mappings under Brjuno-Rüssmann's non-resonant condition, J. Dynam. Differ. Equ., 26 (2014), 989-1005.  doi: 10.1007/s10884-014-9366-1.

[46]

D. ZhangJ. Xu and H. Wu, On Invariant Tori with Prescribed Frequency in Hamiltonian Systems, Adv. Nonlinear Stud., 16 (2016), 719-737.  doi: 10.1515/ans-2015-5051.

[47]

D. Zhang and J. Liang, On high dimensional Schrödinger equation with quasi-periodic potentials, J. Dyn. Control Syst., 23 (2017), 655-666.  doi: 10.1007/s10883-016-9347-2.

[48]

D. ZhangJ. Xu and X. Xu, Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies, Discrete Contin. Dyn. Syst., 38 (2018), 2851-2877.  doi: 10.3934/dcds.2018123.

[49]

H. Zhao, A note on quasi-periodic perturbations of elliptic equilibrium points, Bull. Korean Math. Soc., 49 (2012), 1223-1240.  doi: 10.4134/BKMS.2012.49.6.1223.

[50]

M. Zhang and W. Li, A Lyapunov-type stability criterion using $L^{\alpha}$ norms, Proc. Amer. Math. Soc., 130 (2002), 3325-3333.  doi: 10.1090/S0002-9939-02-06462-6.

[51]

Q. Zhou and J. Wang, Reducibility results for quasiperiodic cocycles with Liouvillean frequency, J. Dynam. Differ. Equ., 24 (2012), 61-83.  doi: 10.1007/s10884-011-9235-0.

[52]

D. Zhang and J. Xu, On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition, Discrete Contin. Dyn. Syst., 16 (2006), 635-655.  doi: 10.3934/dcds.2006.16.635.

[53]

D. Zhang and J. Xu, Invariant tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition, Nonlinear Anal., 67 (2007), 2240-2257.  doi: 10.1016/j.na.2006.09.012.

show all references

References:
[1]

A. AvilaB. Fayad and R. Krikorian, A KAM scheme for SL(2, R) cocycles with Liouvillean frequencies, Geom. Funct. Anal., 21 (2011), 1001-1019.  doi: 10.1007/s00039-011-0135-6.

[2]

D. BambusiB. GrébertA. Maspero and D. Robert, Reducibility of the quantum harmonic oscillator in d-dimensions with polynomial time-dependent perturbation, Anal. Partial Differ. Equ., 11 (2018), 775-799.  doi: 10.2140/apde.2018.11.775.

[3]

D. Bambusi, Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations, I, Trans. Amer. Math. Soc., 370 (2018), 1823-1865.  doi: 10.1090/tran/7135.

[4]

D. Bambusi, Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations, II, Comm. Math. Phys., 353 (2017), 353-378.  doi: 10.1007/s00220-016-2825-2.

[5]

Yu. N. Bibikov, On the stability of the zero solution of essentially nonlinear Hamiltonian systems and reversible systems with one degree of freedom, Differ. Equ., 38 (2002), 609-614.  doi: 10.1023/A:1020298221798.

[6]

A. Bounemoura, Effective stability for Gevrey and finitely differentiable prevalent Hamiltonians, Commun. Math. Phys., 307 (2011), 157-183.  doi: 10.1007/s00220-011-1306-x.

[7]

A. Bounemoura and J. Féjoz, KAM, $\alpha$-Gevrey regularity and the $\alpha$-Bruno-Rüssmann condition, Ann. Sc. Norm. Super. Pisa Cl. Sci., 19 (2019), 1225-1279. 

[8]

C. Chavaudret and S. Marmi, Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition, J. Mod. Dyn., 6 (2012), 59-78.  doi: 10.3934/jmd.2012.6.59.

[9]

C. Chavaudret, Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles, Bull. Soc. Math. France, 141 (2013), 47-106.  doi: 10.24033/bsmf.2643.

[10]

C. Chavaudret and L. Stolovitch, Analytic reducibility of resonant cocycles to a normal form, J. Inst. Math. Jussieu, 15 (2016), 203-223.  doi: 10.1017/S1474748014000383.

[11]

E. I. Dinaburg and Ya. G. Sinai, The one dimensional Schrödinger equation with quasi-perioidc potential, Funct. Anal. Appl., 9 (1975), 8-21. 

[12]

L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Commun. Math. Phys., 146 (1992), 447-482. 

[13]

L. H. Eliasson, Almost reducibility of linear quasi-periodic systems, Smooth ergodic theory and its applications (Seattle, WA, 1999), 679–705, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001. doi: 10.1090/pspum/069/1858550.

[14]

L. H. Eliasson and S. B. Kuksin, On reducibility of Schrödinger equations with quasi-periodic in time potentials, Commun. Math. Phys., 286 (2009), 125-135.  doi: 10.1007/s00220-008-0683-2.

[15]

B. Fayad and R. Krikorian, Herman's last geometric theorem, Ann. Sci. Éc. Norm. Supér., 42 (2009), 193–219. doi: 10.24033/asens.2093.

[16]

H. Her and J. You, Full measure reducibility for generic one-parameter family of quasi-periodic linear systems, J. Dynam. Differ. Equ., 20 (2008), 831-866.  doi: 10.1007/s10884-008-9113-6.

[17]

X. Hou and J. You, Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems, Invent. Math., 190 (2012), 209-260.  doi: 10.1007/s00222-012-0379-2.

[18]

R. A. Johnson and G. R. Sell, Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems, J. Differ. Equ., 41 (1981), 262-288.  doi: 10.1016/0022-0396(81)90062-0.

[19]

À. Jorba and C. Simó, On the reducibility of linear differential equation with quasi-perioidc coefficients, J. Differ. Equ., 98 (1992), 111-124.  doi: 10.1016/0022-0396(92)90107-X.

[20]

À. Jorba and C. Simó, On quasi-periodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal., 27 (1996), 1704-1737.  doi: 10.1137/S0036141094276913.

[21]

R. Krikorian, Global density of reducible quasi-periodic cocycles on $\mathbb{T}^{1} \times SU(2)$, Ann. Math., 154 (2001), 269-326.  doi: 10.2307/3062098.

[22]

B. Liu, The stability of the equilibrium of planar Hamiltonian and reversible systems, J. Dynam. Differ. Equ., 18 (2006), 975-990.  doi: 10.1007/s10884-006-9027-0.

[23]

B. Liu, The stability of equilibrium of quasi-periodic planar Hamiltonian and reversible systems, Sci. China Math., 53 (2010), 125-136.  doi: 10.1007/s11425-009-0117-4.

[24]

J. Lopes Dias, A normal form theorem for Brjuno skew systems through renormalization, J. Differ. Equ., 230 (2006), 1-23.  doi: 10.1016/j.jde.2006.07.021.

[25]

J. Lopes Dias, Brjuno condition and renormalization for Poincaré flows, Discrete Contin. Dyn. Syst., 15 (2006), 641-656.  doi: 10.3934/dcds.2006.15.641.

[26]

J. Lopes Dias and J. Pedro Gaivão, Linearization of Gevrey flows on $\mathbb{T}^{d}$ with a Brjuno type arithmetical condition, J. Differ. Equ., 267 (2019), 7167-7212.  doi: 10.1016/j.jde.2019.07.020.

[27]

J. Liang and J. Xu, A note on the extension of Dinaburg-Sinai theorem to higher dimension, Ergodic Theory Dynam. Syst., 25 (2005), 1539-1549.  doi: 10.1017/S0143385705000118.

[28]

W. Magnus and S. Winkler, Hill's Equation, Corrected reprint of the 1966 edition. Dover Publications, Inc., New York, 1979.

[29]

J. P. Marco and D. Sauzin, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems, Publ. Math. Inst. Hautes Études Sci., No. 96 (2002), 199–275 (2003). doi: 10.1007/s10240-003-0011-5.

[30]

J. Pöschel, KAM $\grave{a}$ la R, Regul. Chaotic Dyn., 16 (2011), 17-23.  doi: 10.1134/S1560354710520060.

[31]

G. Popov, KAM theorem for Gevrey Hamiltonians, Ergodic Theory Dynam. Systems, 24 (2004), 1753-1786.  doi: 10.1017/S0143385704000458.

[32]

H. Rüssmann, On the one dimensional Schrödinger equation with a quasi-perioidc potential, Ann. New York Acad. Sci., 357 (1980), 90-107. 

[33]

H. Rüssmann, Convergent transformations into a normal form in analytic Hamiltonian systems with two degrees of freedom on the zero energy surface near degenerate elliptic singularities, Ergodic Theory Dynam. Systems, 24 (2004), 1787-1832.  doi: 10.1017/S0143385703000774.

[34]

J. WangJ. You and Q. Zhou, Response solutions for quasi-periodically forced harmonic oscillators, Trans. Amer. Math. Soc., 369 (2017), 4251-4274.  doi: 10.1090/tran/6800.

[35]

X. Wang and J. Xu, On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point, Nonlinear Anal., 69 (2008), 2318-2329.  doi: 10.1016/j.na.2007.08.016.

[36]

X. WangJ. Xu and D. Zhang, On the persistence of degenerate lower-dimensional tori in reversible systems, Ergodic Theory Dynam. Systems, 35 (2015), 2311-2333.  doi: 10.1017/etds.2014.34.

[37]

Y. Wu and Y. Wang, The stability of the elliptic equilibuium of planar quasi-periodic Hamiltonian system, Acta Math. Sin., 28 (2002), 801-816.  doi: 10.1007/s10114-011-0006-y.

[38]

Zh. Wang and Zh. Liang, Reducibility of 1D quantum harmonic oscillator perturbed by a quasiperiodic potential with logarithmic decay, Nonlinearity, 30 (2017), 1405-1448.  doi: 10.1088/1361-6544/aa5d6c.

[39]

J. Xu and Q. Zheng, On the reducibility of linear differential equations with quasiperiodic coefficients which are degenerate, Proc. Amer. Math. Soc., 126 (1998), 1445-1451.  doi: 10.1090/S0002-9939-98-04523-7.

[40]

J. Xu, On the reducibility of a class of linear differential equation with quasi-periodic coefficients, Mathematika, 46 (1999), 443-451.  doi: 10.1112/S0025579300007907.

[41]

J. Xu and J. You, Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition, J. Math. Pures Appl., 80 (2001), 1045-1067.  doi: 10.1016/S0021-7824(01)01221-1.

[42]

J. Xu, Normal form of reversible systems and persistence of lower dimensional tori under weaker nonresonance conditions, SIAM J. Math. Anal., 36 (2004), 233-255.  doi: 10.1137/S0036141003421923.

[43]

J. Xu and X. Lu, On the reducibility of two-dimensional linear quasi-periodic systems with small parameter, Ergodic Theory Dynam. Systems, 35 (2015), 2334-2352.  doi: 10.1017/etds.2014.31.

[44]

X. Yuan and A. Nunes, A note on the reducibility of linear differential equations with quasiperiodic coefficients, Int. J. Math. Math. Sci., 2003, 4071–4083. doi: 10.1155/S0161171203206025.

[45]

D. Zhang and J. Xu, Invariant curves of analytic reversible mappings under Brjuno-Rüssmann's non-resonant condition, J. Dynam. Differ. Equ., 26 (2014), 989-1005.  doi: 10.1007/s10884-014-9366-1.

[46]

D. ZhangJ. Xu and H. Wu, On Invariant Tori with Prescribed Frequency in Hamiltonian Systems, Adv. Nonlinear Stud., 16 (2016), 719-737.  doi: 10.1515/ans-2015-5051.

[47]

D. Zhang and J. Liang, On high dimensional Schrödinger equation with quasi-periodic potentials, J. Dyn. Control Syst., 23 (2017), 655-666.  doi: 10.1007/s10883-016-9347-2.

[48]

D. ZhangJ. Xu and X. Xu, Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies, Discrete Contin. Dyn. Syst., 38 (2018), 2851-2877.  doi: 10.3934/dcds.2018123.

[49]

H. Zhao, A note on quasi-periodic perturbations of elliptic equilibrium points, Bull. Korean Math. Soc., 49 (2012), 1223-1240.  doi: 10.4134/BKMS.2012.49.6.1223.

[50]

M. Zhang and W. Li, A Lyapunov-type stability criterion using $L^{\alpha}$ norms, Proc. Amer. Math. Soc., 130 (2002), 3325-3333.  doi: 10.1090/S0002-9939-02-06462-6.

[51]

Q. Zhou and J. Wang, Reducibility results for quasiperiodic cocycles with Liouvillean frequency, J. Dynam. Differ. Equ., 24 (2012), 61-83.  doi: 10.1007/s10884-011-9235-0.

[52]

D. Zhang and J. Xu, On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition, Discrete Contin. Dyn. Syst., 16 (2006), 635-655.  doi: 10.3934/dcds.2006.16.635.

[53]

D. Zhang and J. Xu, Invariant tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition, Nonlinear Anal., 67 (2007), 2240-2257.  doi: 10.1016/j.na.2006.09.012.

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