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

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## Abstract

In this paper, we study the reducibility problem for a class of analytic almost periodic linear Hamiltonian systems where

$$\begin{aligned} \frac{dx}{dt} = J[{A}+\varepsilon {Q}(t)]x \end{aligned}$$

*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*).## Keywords

Almost periodic matrix Reducibility KAM iteration Linear Hamiltonian systems## Mathematics Subject Classification

37C10 70H08## Notes

### Acknowledgements

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).

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