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

Xiaocai Wang; Junxiang Xu; Dongfeng Zhang

doi:10.3934/dcds.2017092

Article Contents

2017, Volume 37, Issue 4: 2141-2160. Doi: 10.3934/dcds.2017092

This issue Previous Article Degenerate with respect to parameters fold-Hopf bifurcations Next Article Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients

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

Author Bio: E-mail address: wangxiaocai325@163.com(X.Wang); E-mail address: xujun@seu.edu.cn(J.Xu); E-mail address: zhdf@seu.edu.cn(D.Zhang)

Received: March 31, 2016

Revised: October 31, 2016

Published: May 2017

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).

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Abstract

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].

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Mathematics Subject Classification: Primary:37J40;Secondary:37E99, 47A55, 37F50.

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References

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