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June 2018, 38(6): 2851-2877. doi: 10.3934/dcds.2018123
Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies
Dongfeng Zhang , , Junxiang Xu and Xindong Xu
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)
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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.
Keywords: Skew symmetric systems, reducibility, KAM iteration, Liouvillean frequencies.
Mathematics Subject Classification: Primary: 70H08; Secondary: 37J40.
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
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
D. Bambusi, Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations, Ⅰ, preprint, arXiv: 1606.04494 [math. DS]. Google Scholar
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
J. Pöschel, KAM $\grave{a}$ la R, Regul. Chaotic Dyn., 16 (2011), 17-23. doi: 10.1134/S1560354710520060. Google Scholar
H. Rüssmann, On the one dimensional Schrödinger equation with a quasi-perioidc potential, Ann. New York Acad. Sci., 357 (1980), 90-107. Google Scholar
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
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
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
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
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
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
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
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
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
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
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
D. Zhang, J. Xu and X. Xu, Quasi-periodic solutions of high dimensional Schrödinger equation with Liouvillean basic frequencies, Submitted. Google Scholar
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
Helmut Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 683-718. doi: 10.3934/dcdss.2010.3.683
Saša Kocić. Reducibility of skew-product systems with multidimensional Brjuno base flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 261-283. doi: 10.3934/dcds.2011.29.261
Hans Koch, João Lopes Dias. Renormalization of diophantine skew flows, with applications to the reducibility problem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 477-500. doi: 10.3934/dcds.2008.21.477
Yang Cao, Wei- Wei Tan, Mei-Qun Jiang. A generalization of the positive-definite and skew-Hermitian splitting iteration. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 811-821. doi: 10.3934/naco.2012.2.811
Tahereh Salimi Siahkolaei, Davod Khojasteh Salkuyeh. A preconditioned SSOR iteration method for solving complex symmetric system of linear equations. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 483-492. doi: 10.3934/naco.2019033
Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. On the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1677-1692. doi: 10.3934/dcds.2016.36.1677
Álvaro Castañeda, Gonzalo Robledo. Almost reducibility of linear difference systems from a spectral point of view. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1977-1988. doi: 10.3934/cpaa.2017097
Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2141-2160. doi: 10.3934/dcds.2017092
Patrik Nystedt, Johan Öinert. Simple skew category algebras associated with minimal partially defined dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4157-4171. doi: 10.3934/dcds.2013.33.4157
Renato Manfrin. On the global solvability of symmetric hyperbolic systems of Kirchhoff type. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 91-106. doi: 10.3934/dcds.1997.3.91
Alexandra Skripchenko. Symmetric interval identification systems of order three. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 643-656. doi: 10.3934/dcds.2012.32.643
Inês Cruz, M. Esmeralda Sousa-Dias. Reduction of cluster iteration maps. Journal of Geometric Mechanics, 2014, 6 (3) : 297-318. doi: 10.3934/jgm.2014.6.297
Manil T. Mohan, Sivaguru S. Sritharan. New methods for local solvability of quasilinear symmetric hyperbolic systems. Evolution Equations & Control Theory, 2016, 5 (2) : 273-302. doi: 10.3934/eect.2016005
André Vanderbauwhede. Continuation and bifurcation of multi-symmetric solutions in reversible Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 359-363. doi: 10.3934/dcds.2013.33.359
Chungen Liu. Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 337-355. doi: 10.3934/dcds.2010.27.337
Alessandra Celletti. Some KAM applications to Celestial Mechanics. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 533-544. doi: 10.3934/dcdss.2010.3.533
David Maxwell. Kozlov-Maz'ya iteration as a form of Landweber iteration. Inverse Problems & Imaging, 2014, 8 (2) : 537-560. doi: 10.3934/ipi.2014.8.537
Roy Adler, Bruce Kitchens, Michael Shub. Stably ergodic skew products. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 349-350. doi: 10.3934/dcds.1996.2.349
Jory Griffin, Jens Marklof. Limit theorems for skew translations. Journal of Modern Dynamics, 2014, 8 (2) : 177-189. doi: 10.3934/jmd.2014.8.177
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