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April  2017, 37(4): 2141-2160. doi: 10.3934/dcds.2017092

## A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems

 1 Faculty of mathematics and physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, China 2 Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China

Received  April 2016 Revised  November 2016 Published  December 2016

Fund Project: The first author is supported by National Natural Science Foundation of China (11501234) and Qing Lan Project. The second author is supported by National Natural Science Foundation of China (11371090). The third author is supported by National Natural Science Foundation of China (11001048).

In this paper we consider the persistence of elliptic lower dimensional invariant tori with one normal frequency in reversible systems, andprove that if the frequency mapping
 $ω(y) ∈ \mathbb{R}^n$
and normal frequency mapping
 $λ(y) ∈ \mathbb{R}$
satisfy that
 $\text{deg} (ω/λ ,\mathcal{O},ω_0/λ_0)≠ 0,$
where
 $ω_0 =ω(y_0)$
and
 $λ_0 = λ(y_0)$
satisfy Melnikov's non-resonance conditions for some
 $y_0∈\mathcal{O}$
, then the direction of this frequency for the invariant torus persists under small perturbations. Our result is a generalization of X. Wang et al[Persistence of lower dimensional elliptic invariant tori for a class of nearly integrablereversible systems, Discrete and Continuous Dynamical Systems series B, 14 (2010), 1237-1249].
Citation: 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
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