Analytic integrable systems: Analytic normalization and embedding flows
Analytic integrable systems: Analytic normalization and embedding flows
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DOI：
10.1016/j.jde.2013.01.016
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发表时间：
2014-07
期刊：
影响因子：
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通讯作者：
Zhang Xiang
中科院分区：
文献类型：
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作者：
Zhang Xiang
本文主要研究了有限维完全解析可积动力系统解析归一化的存在性和范式。更详细地，我们将证明在（Cn,0）中任何完全解析可积微分同态F(x)=Bx+ F(x)，且B的特征值不是模1，且F(x)= O（|x|2）是局部解析共轭到其范式的。同时，我们也证明了在（Cn,0）中，当A具有非零特征值且f(x)=O（|x|2）时，任何完全解析可积微分系统x˙=Ax+f(x)是局部解析共轭到其范式的。进一步证明了定义在解析流形上的任何完全解析可积微分同态可以嵌入到完全解析可积流中。我们注意到，我们的部分结果是对Moser在J. Moser， the analytic invariants of a area-preserving mapping near a双曲不动点，Comm. Pure applet中的结果的改进。数学，9(1956)673-692和庞卡罗莱在H.庞卡罗莱，Sur l ' intacgration des samquations diff， II, Rend。约巴勒莫11(1897)193-239。这些结果也改进了张xiang，解析可积系统的解析归一化和嵌入流，J.微分方程244(2008)1080-1092中的结果，即系统的线性部分可以是非双曲的，以及N.T. Zung, poincar<s:1> - dulac范式中的收敛性与可积性，数学。参考编9(2002)217-228，在我们的论文中给出了在限制情况下范式的具体表达式。
In this paper we mainly study the existence of analytic normalization and the normal form of finite dimensional complete analytic integrable dynamical systems. More details, we will prove that any complete analytic integrable diffeomorphism F(x)=Bx+f(x) in (Cn,0) with B having eigenvalues not modulus 1 and f(x)=O(|x|2) is locally analytically conjugate to its normal form. Meanwhile, we also prove that any complete analytic integrable differential system x˙=Ax+f(x) in (Cn,0) with A having nonzero eigenvalues and f(x)=O(|x|2) is locally analytically conjugate to its normal form. Furthermore we will prove that any complete analytic integrable diffeomorphism defined on an analytic manifold can be embedded in a complete analytic integrable flow. We note that parts of our results are the improvement of Moserʼs one in J. Moser, The analytic invariants of an area-preserving mapping near a hyperbolic fixed point, Comm. Pure Appl. Math. 9 (1956) 673–692 and of Poincaréʼs one in H. Poincaré, Sur lʼintégration des équations différentielles du premier order et du premier degré, II, Rend. Circ. Mat. Palermo 11 (1897) 193–239. These results also improve the ones in Xiang Zhang, Analytic normalization of analytic integrable systems and the embedding flows, J. Differential Equations 244 (2008) 1080–1092 in the sense that the linear part of the systems can be nonhyperbolic, and the one in N.T. Zung, Convergence versus integrability in Poincaré–Dulac normal form, Math. Res. Lett. 9 (2002) 217–228 in the way that our paper presents the concrete expression of the normal form in a restricted case.