# American Institute of Mathematical Sciences

June  2018, 38(6): 2851-2877. doi: 10.3934/dcds.2018123

## Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies

 School of Mathematics, Southeast University, Nanjing 210096, China

* Corresponding author: Dongfeng Zhang

Received  July 2017 Revised  December 2017 Published  April 2018

Fund Project: The first author is supported by NSF grant (11001048) (11571072) and the Fundamental Research Funds for the Central Universities (2242017k41046), The second author is supported by NSF grant (11371090), The third author is supported by NSF grant (11771077)

In this paper we consider the system $\dot{x} = (A(\epsilon)+ \epsilon^{m} P(t;\epsilon)) x, x∈\mathbb{R}^{3},$ where $\epsilon$ is a small parameter, $A, P$ are all $3×3$ skew symmetric matrices, $A$ is a constant matrix with eigenvalues $± i\bar{λ}(\epsilon)$ and 0, where $\bar{λ}(\epsilon) = λ+a_{m_{0}}\epsilon^{m_{0}} + O(\epsilon^{m_{0}+1}) (m_{0}< m),$ $a_{m_{0}}≠ 0,$ $P$ 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.

Citation: Dongfeng Zhang, Junxiang Xu, Xindong Xu. Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2851-2877. doi: 10.3934/dcds.2018123
##### References:
 [1] A. Avila, B. 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. Google Scholar [2] D. Bambusi, Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations, Ⅰ, preprint, arXiv: 1606.04494 [math. DS].Google Scholar [3] 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. Google Scholar [4] 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. Google Scholar [5] 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. Google Scholar [6] 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. Google Scholar [7] E. I. Dinaburg and Ya. G. Sinai, The one dimensional Schrödinger equation with quasi-perioidc potential, Funct. Anal. Appl., 9 (1975), 8-21. Google Scholar [8] L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys., 146 (1992), 447-482. doi: 10.1007/BF02097013. Google Scholar [9] 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. Google Scholar [10] L. H. Eliasson, Ergodic skew-systems on $\mathbb{T}^{d} \times SO(3, \mathbb{R})$, Ergodic Theory Dynam. Systems, 22 (2002), 1429-1449. Google Scholar [11] L. H. Eliasson and S. B. Kuksin, On reducibility of Schrödinger equations with quasi-periodic in time potentials, Comm. Math. Phys., 286 (2009), 125-135. doi: 10.1007/s00220-008-0683-2. Google Scholar [12] B. Fayad and R. Krikorian, Herman's last geometric theorem, Ann. Sci. Éc. Norm. Supér., 42 (2009), 193-219. doi: 10.24033/asens.2093. Google Scholar [13] H. Her and J. You, Full measure reducibility for generic one-parameter family of quasi-periodic linear systems, J. Dynam. Differential Equations, 20 (2008), 831-866. doi: 10.1007/s10884-008-9113-6. Google Scholar [14] 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. Google Scholar [15] R. A. Johnson and G. R. Sell, Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems, J. Differential Equations, 41 (1981), 262-288. doi: 10.1016/0022-0396(81)90062-0. Google Scholar [16] À. Jorba and C. Simó, On the reducibility of linear differential equation with quasi-perioidc coefficients, J. Differential Equations, 98 (1992), 111-124. doi: 10.1016/0022-0396(92)90107-X. Google Scholar [17] 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. Google Scholar [18] R. Krikorian, Global density of reducible quasi-periodic cocycles on $T^{1} \times SU(2)$, Ann. of Math., 154 (2001), 269-326. doi: 10.2307/3062098. Google Scholar [19] J. Liang and J. Xu, A note on the extension of Dinaburg-Sinai theorem to higher dimension, Ergodic Theory Dynam. Systems, 25 (2005), 1539-1549. doi: 10.1017/S0143385705000118. Google Scholar [20] J. Lopes Dias, A normal form theorem for Brjuno skew systems through renormalization, J. Differential Equations, 230 (2006), 1-23. doi: 10.1016/j.jde.2006.07.021. Google Scholar [21] 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. Google Scholar [22] J. Pöschel, KAM $\grave{a}$ la R, Regul. Chaotic Dyn., 16 (2011), 17-23. doi: 10.1134/S1560354710520060. Google Scholar [23] H. Rüssmann, On the one dimensional Schrödinger equation with a quasi-perioidc potential, Ann. New York Acad. Sci., 357 (1980), 90-107. Google Scholar [24] 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. Google Scholar [25] J. Wang, J. You and Q. Zhou, Response solutions for quasi-periodically forced harmonic oscillators, Trans. Amer. Math. Soc., 369 (2017), 4251-4274. Google Scholar [26] 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. Google Scholar [27] X. Wang, J. 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. Google Scholar [28] 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. Google Scholar [29] 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. Google Scholar [30] 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. Google Scholar [31] D. Zhang and J. Xu, Invariant curves of analytic reversible mappings under Brjuno-Rüssmann's non-resonant condition, J. Dynam. Differential Equations, 26 (2014), 989-1005. doi: 10.1007/s10884-014-9366-1. Google Scholar [32] D. Zhang and J. Xu, On invariant tori of vector field under weaker non-degeneracy condition, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1381-1394. doi: 10.1007/s00030-015-0326-1. Google Scholar [33] D. Zhang, J. Xu and H. Wu, On invariant tori with prescribed frequency in Hamiltonian systems, Advanced Nonlinear Studies, 16 (2016), 719-735. Google Scholar [34] 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. Google Scholar [35] D. Zhang, J. Xu and X. Xu, Quasi-periodic solutions of high dimensional Schrödinger equation with Liouvillean basic frequencies, Submitted.Google Scholar [36] Q. Zhou and J. Wang, Reducibility results for quasiperiodic cocycles with Liouvillean frequency, J. Dynam. Differential Equations, 24 (2012), 61-83. doi: 10.1007/s10884-011-9235-0. Google Scholar

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##### References:
 [1] A. Avila, B. 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. Google Scholar [2] D. Bambusi, Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations, Ⅰ, preprint, arXiv: 1606.04494 [math. DS].Google Scholar [3] 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. Google Scholar [4] 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. Google Scholar [5] 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. Google Scholar [6] 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. Google Scholar [7] E. I. Dinaburg and Ya. G. Sinai, The one dimensional Schrödinger equation with quasi-perioidc potential, Funct. Anal. Appl., 9 (1975), 8-21. Google Scholar [8] L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys., 146 (1992), 447-482. doi: 10.1007/BF02097013. Google Scholar [9] 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. Google Scholar [10] L. H. Eliasson, Ergodic skew-systems on $\mathbb{T}^{d} \times SO(3, \mathbb{R})$, Ergodic Theory Dynam. Systems, 22 (2002), 1429-1449. Google Scholar [11] L. H. Eliasson and S. B. Kuksin, On reducibility of Schrödinger equations with quasi-periodic in time potentials, Comm. Math. Phys., 286 (2009), 125-135. doi: 10.1007/s00220-008-0683-2. Google Scholar [12] B. Fayad and R. Krikorian, Herman's last geometric theorem, Ann. Sci. Éc. Norm. Supér., 42 (2009), 193-219. doi: 10.24033/asens.2093. Google Scholar [13] H. Her and J. You, Full measure reducibility for generic one-parameter family of quasi-periodic linear systems, J. Dynam. Differential Equations, 20 (2008), 831-866. doi: 10.1007/s10884-008-9113-6. Google Scholar [14] 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. Google Scholar [15] R. A. Johnson and G. R. Sell, Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems, J. Differential Equations, 41 (1981), 262-288. doi: 10.1016/0022-0396(81)90062-0. Google Scholar [16] À. Jorba and C. Simó, On the reducibility of linear differential equation with quasi-perioidc coefficients, J. Differential Equations, 98 (1992), 111-124. doi: 10.1016/0022-0396(92)90107-X. Google Scholar [17] 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. Google Scholar [18] R. Krikorian, Global density of reducible quasi-periodic cocycles on $T^{1} \times SU(2)$, Ann. of Math., 154 (2001), 269-326. doi: 10.2307/3062098. Google Scholar [19] J. Liang and J. Xu, A note on the extension of Dinaburg-Sinai theorem to higher dimension, Ergodic Theory Dynam. Systems, 25 (2005), 1539-1549. doi: 10.1017/S0143385705000118. Google Scholar [20] J. Lopes Dias, A normal form theorem for Brjuno skew systems through renormalization, J. Differential Equations, 230 (2006), 1-23. doi: 10.1016/j.jde.2006.07.021. Google Scholar [21] 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. Google Scholar [22] J. Pöschel, KAM $\grave{a}$ la R, Regul. Chaotic Dyn., 16 (2011), 17-23. doi: 10.1134/S1560354710520060. Google Scholar [23] H. Rüssmann, On the one dimensional Schrödinger equation with a quasi-perioidc potential, Ann. New York Acad. Sci., 357 (1980), 90-107. Google Scholar [24] 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. Google Scholar [25] J. Wang, J. You and Q. Zhou, Response solutions for quasi-periodically forced harmonic oscillators, Trans. Amer. Math. Soc., 369 (2017), 4251-4274. Google Scholar [26] 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. Google Scholar [27] X. Wang, J. 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. Google Scholar [28] 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. Google Scholar [29] 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. Google Scholar [30] 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. Google Scholar [31] D. Zhang and J. Xu, Invariant curves of analytic reversible mappings under Brjuno-Rüssmann's non-resonant condition, J. Dynam. Differential Equations, 26 (2014), 989-1005. doi: 10.1007/s10884-014-9366-1. Google Scholar [32] D. Zhang and J. Xu, On invariant tori of vector field under weaker non-degeneracy condition, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1381-1394. doi: 10.1007/s00030-015-0326-1. Google Scholar [33] D. Zhang, J. Xu and H. Wu, On invariant tori with prescribed frequency in Hamiltonian systems, Advanced Nonlinear Studies, 16 (2016), 719-735. Google Scholar [34] 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. Google Scholar [35] D. Zhang, J. Xu and X. Xu, Quasi-periodic solutions of high dimensional Schrödinger equation with Liouvillean basic frequencies, Submitted.Google Scholar [36] Q. Zhou and J. Wang, Reducibility results for quasiperiodic cocycles with Liouvillean frequency, J. Dynam. Differential Equations, 24 (2012), 61-83. doi: 10.1007/s10884-011-9235-0. Google Scholar
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