# 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 and Continuous Dynamical Systems, 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. [2] D. Bambusi, Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations, Ⅰ, preprint, arXiv: 1606.04494 [math. DS]. [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. [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. [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. [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. [7] E. I. Dinaburg and Ya. G. Sinai, The one dimensional Schrödinger equation with quasi-perioidc potential, Funct. Anal. Appl., 9 (1975), 8-21. [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. [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. [10] L. H. Eliasson, Ergodic skew-systems on $\mathbb{T}^{d} \times SO(3, \mathbb{R})$, Ergodic Theory Dynam. Systems, 22 (2002), 1429-1449. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [22] J. Pöschel, KAM $\grave{a}$ la R, Regul. Chaotic Dyn., 16 (2011), 17-23.  doi: 10.1134/S1560354710520060. [23] H. Rüssmann, On the one dimensional Schrödinger equation with a quasi-perioidc potential, Ann. New York Acad. Sci., 357 (1980), 90-107. [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. [25] J. Wang, J. You and Q. Zhou, Response solutions for quasi-periodically forced harmonic oscillators, Trans. Amer. Math. Soc., 369 (2017), 4251-4274. [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. [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. [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. [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. [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. [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. [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. [33] D. Zhang, J. Xu and H. Wu, On invariant tori with prescribed frequency in Hamiltonian systems, Advanced Nonlinear Studies, 16 (2016), 719-735. [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. [35] D. Zhang, J. Xu and X. Xu, Quasi-periodic solutions of high dimensional Schrödinger equation with Liouvillean basic frequencies, Submitted. [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.

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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. [2] D. Bambusi, Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations, Ⅰ, preprint, arXiv: 1606.04494 [math. DS]. [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. [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. [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. [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. [7] E. I. Dinaburg and Ya. G. Sinai, The one dimensional Schrödinger equation with quasi-perioidc potential, Funct. Anal. Appl., 9 (1975), 8-21. [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. [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. [10] L. H. Eliasson, Ergodic skew-systems on $\mathbb{T}^{d} \times SO(3, \mathbb{R})$, Ergodic Theory Dynam. Systems, 22 (2002), 1429-1449. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [22] J. Pöschel, KAM $\grave{a}$ la R, Regul. Chaotic Dyn., 16 (2011), 17-23.  doi: 10.1134/S1560354710520060. [23] H. Rüssmann, On the one dimensional Schrödinger equation with a quasi-perioidc potential, Ann. New York Acad. Sci., 357 (1980), 90-107. [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. [25] J. Wang, J. You and Q. Zhou, Response solutions for quasi-periodically forced harmonic oscillators, Trans. Amer. Math. Soc., 369 (2017), 4251-4274. [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. [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. [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. [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. [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. [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. [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. [33] D. Zhang, J. Xu and H. Wu, On invariant tori with prescribed frequency in Hamiltonian systems, Advanced Nonlinear Studies, 16 (2016), 719-735. [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. [35] D. Zhang, J. Xu and X. Xu, Quasi-periodic solutions of high dimensional Schrödinger equation with Liouvillean basic frequencies, Submitted. [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.
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