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Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies
School of Mathematics, Southeast University, Nanjing 210096, China |
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.
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. |
show all references
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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