Advanced Search
Article Contents
Article Contents

Reducibility of quasi-periodically forced circle flows

  • * Corresponding author: Saša Kocić

    * Corresponding author: Saša Kocić 
Abstract Full Text(HTML) Related Papers Cited by
  • We develop a renormalization group approach to the problem of reducibility of quasi-periodically forced circle flows. We apply the method to prove a reducibility theorem for such flows.

    Mathematics Subject Classification: Primary: 37E10, 37F25.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] V. I. Arnol'd, Small denominators. Ⅰ: On the mapping of a circle into itself, Izv. Akad. Nauk. Math. Serie, 25 (1961), 21-86. 
    [2] V. I. Arnol'd, Proof of A. N. Kolmogorov's theorem on the preservation of quasiperiodic motions under small perturbations of the Hamiltonian, Uspehi Mat. Nauk, 18 (1963), 13-40. 
    [3] A. D. Brjuno, Analytic form of differential equations. Ⅰ, Trudy Moskov. Mat. Obšč., 25 (1971), 119-262.
    [4] A. D. Brjuno, Analytic form of differential equations. Ⅱ, Trudy Moskov. Mat. Obšč., 26 (1972), 199-239.
    [5] L. Corsi and G. Gentile, Oscillator synchronisation under arbitrary quasi-periodic forcing, Commun. Math. Phys., 316 (2012), 489-529.  doi: 10.1007/s00220-012-1548-2.
    [6] G. Gentile, Resummation of perturbation series and reducibility for Bryuno skew-product flows, J. Stat. Phys., 125 (2006), 321-361.  doi: 10.1007/s10955-006-9127-6.
    [7] G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory and Dynam. Systems, 27 (2007), 427-457.  doi: 10.1017/S0143385706000757.
    [8] M.-R. Herman, Sur la conjugasion differentiable des difféomorphismes du cercle a de rotations, Inst. Hautes études Sci. Publ. Math., 49 (1979), 5-233. 
    [9] R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438.  doi: 10.1007/BF01208484.
    [10] K. KhaninJ. Lopes Dias and J. Marklof, Multidimensional continued fractions, dynamic renormalization and KAM theory, Commun. Math. Phys., 270 (2007), 197-231.  doi: 10.1007/s00220-006-0125-y.
    [11] H. Koch and S. Kocić, Renormalization of vector fields and Diophantine invariant tori, Ergodic Theory Dynam. Systems, 28 (2008), 1559-1585.  doi: 10.1017/S0143385707000892.
    [12] H. Koch and S. Kocić, A renormalization group approach to quasiperiodic motion with Brjuno frequencies, Ergodic Theory Dynam. Systems, 30 (2010), 1131-1146.  doi: 10.1017/S014338570900042X.
    [13] H. Koch and S. Kocić, A renormalization approach to lower-dimensional tori with Brjuno frequency vectors, J. Differential Equations, 249 (2010), 1986-2004.  doi: 10.1016/j.jde.2010.05.004.
    [14] S. Kocić, Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori, Nonlinearity, 18 (2005), 2513-2544.  doi: 10.1088/0951-7715/18/6/006.
    [15] S. Kocić, Reducibility of skew-product systems with multidimensional Brjuno base flows, Discrete Contin. Dyn. Syst. A, 29 (2011), 261-283.  doi: 10.3934/dcds.2011.29.261.
    [16] A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530. 
    [17] 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.
    [18] J. Lopes Dias, Local conjugacy classes for analytic torus flows, J. Differential Equations, 245 (2008), 468-489.  doi: 10.1016/j.jde.2008.04.006.
    [19] J. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176.  doi: 10.1007/BF01399536.
    [20] H. Rüssmann, On the one-dimensional Schrödinger equation with a quasiperiodic potential, Nonlinear Dynamics, Ann. New York Acad. Sci., New York Acad. Sci., New York, 357 (1980), 90-107. 
    [21] J.-C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, 231 (1995).
    [22] J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms, Dynamical Systems and Small Divisors, Lecture Notes in Mathematics, Fond. CIME/CIME Found. Subser., Springer, Berlin, 1784 (2002), 125-173.  doi: 10.1007/978-3-540-47928-4_3.
  • 加载中

Article Metrics

HTML views(215) PDF downloads(338) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint