On the Reducibility of a Class of Linear Almost Periodic Hamiltonian Systems

  • Muhammad Afzal
  • Shuzheng Guo
  • Daxiong PiaoEmail author


In this paper, we study the reducibility problem for a class of analytic almost periodic linear Hamiltonian systems
$$\begin{aligned} \frac{dx}{dt} = J[{A}+\varepsilon {Q}(t)]x \end{aligned}$$
where A is a symmetric matrix, J is an anti-symmetric symplectic matrix, Q(t) is an analytic almost periodic symmetric matrix with respect to t, and \(\varepsilon \) is a sufficiently small parameter. It is also assumed that JA has possible multiple eigenvalues and the basic frequencies of Q satisfy the non-resonance conditions. It is shown that, under some non-resonant conditions, some non-degeneracy conditions and for most sufficiently small \(\varepsilon \) , the Hamiltonian system can be reduced to a constant coefficients Hamiltonian system by means of an almost periodic symplectic change of variables with the same basic frequencies as Q(t).


Almost periodic matrix Reducibility KAM iteration Linear Hamiltonian systems 

Mathematics Subject Classification

37C10 70H08 



The authors are very grateful to the referees for the valuable comments and suggestions. The authors were supported by the NSFC (grant no. 11571327), NSF of Shandong Province (grant no. ZR2013AM026) and the first author was supported by Chinese Scholarship Council (CSC No. 2014GXY552).


  1. 1.
    Fink, A.M.: Almost Periodic Differential Equations, Lecture Notes in Mathematics, vol. 377. Springer, Berlin, Germany (1974)CrossRefGoogle Scholar
  2. 2.
    Jorba, À., Simó, C.: On the reducibility of linear differential equations with quasiperiodic coefficients. J. Differ. Equ. 98(1), 111–124 (1992)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Jorba, À., Simó, C.: On quasi-periodic perturbations of elliptic equilibrium points. SIAM J. Math. Anal. 27, 1704–1737 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Meyer, K.R., Hall, G.R.: Introduction to Hamiltonian Dynamical Systems and the \(n\)-Body Problem, pp. 82–90. Springer, New York (1992)CrossRefGoogle Scholar
  5. 5.
    Palmer, K.J.: On the reducibility of almost periodic systems of linear differential equations. J. Differ. Equ. 36, 374–390 (1980)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Pöschel, J.: Small divisors with spatial structure in infinite dimensional Hamiltonian systems. Commun. Math. Phys. 127(2), 351–393 (1990)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Rüssmann, H.: On the one-dimensional Schrödinger equation with a quasi-periodic potential. Ann. N. Y. Acad. Sci. 357(1), 90–107 (1980)CrossRefGoogle Scholar
  8. 8.
    Xu, J.: On the reducibility of a class of linear differential equations with quasiperiodic coefficients. Mathematika 46(2), 443–451 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Xu, J., You, J.: On the reducibility of linear differential equations with almost periodic coefficients. Chin. Ann. Math. A 17(5), 607–616 (1996). (in Chinese) MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesOcean University of ChinaQingdaoPeople’s Republic of China

Personalised recommendations