Reducibility of a class of 2k-dimensional Hamiltonian systems with quasi-periodic coefficients

ABSTRACT

In this paper, we consider the following real analytic Hamiltonian system $\dot{x}=(A+\epsilon Q(t,\epsilon \left)\right)x,x\in R^{2k},$ where A is a constant Hamiltonian matrix with the different eigenvalues $\pm w_1\sqrt{-1},$ $\pm {\lambda }_2,\dots ,\pm {\lambda }_k$, where $w_1\in R,$ $\pm {\lambda }_i\ne 0$ for $2\le i\le k$ are real, and $Q(t,\epsilon )$ is quasi-periodic with frequencies $w_1,w_2,\dots w_r$. Without any non-degeneracy condition with respect to ϵ, we prove that by a quasi-periodic symplectic mapping, then for most of the sufficiently small parameter ϵ, the Hamiltonian system is reducible.

KEYWORDS:

• Reducibility
• Hamiltonian systems
• quasi-periodic

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first and third authors are supported by National Natural Science Foundation of China (NSFC) grant 11801492 and 11526177 (Tianyuan Foundation), the Natural Science Foundations for Colleges and Universitiesin Jiangsu Province grant 18KJB110029, and the Scientific Research Foundation of Xuzhou Institute of Technology grant XKY2016215. The second author is supported by Natural Science Foundation of China grant 11501560, the Natural Science Foundation of JiangSu Province grant BK20151160, and 333 High-Level Talents Training Program of Jiangsu Province grant BRA2016275.