Dulac maps of real saddle-nodes

IF 1.6 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity Pub Date : 2024-09-17 DOI:10.1088/1361-6544/ad76f3

Yu Ilyashenko

引用次数: 0

Abstract

Consider a germ of a holomorphic vector field at the origin on the coordinate complex plane. This germ is called a saddle-node if the origin is its singular point, one of its eigenvalues at zero is zero, and the other is not. A saddle-node germ is real if its restriction to the real plane is real. The monodromy transformation for this germ has a multiplier at zero equal to 1. The germ of this map is parabolic and admits a ‘normalizing cochain’. In this note we express the Dulac map of any real saddle-node up to a left composition with a real germ through one component of the cochain normalizing the monodromy transformation.

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实鞍节点的杜拉克映射

考虑坐标复平面上原点处全形向量场的一个胚芽。如果原点是它的奇异点，它在零点的一个特征值为零，而另一个不为零，那么这个胚称为鞍节点。如果鞍节点胚芽对实数平面的限制是实数，那么它就是实数胚芽。该胚芽的单旋转变换在零点的乘数等于 1。该映射的胚芽是抛物线形的，并允许 "归一化共链"。在本注释中，我们将通过对单色变换进行归一化处理的共链的一个分量来表达任何实鞍节点的杜拉克映射，直到它与实胚芽的左合成。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Nonlinearity 物理-物理：数学物理

CiteScore

3.00

自引率

5.90%

发文量

170

审稿时长

12 months

期刊介绍： Aimed primarily at mathematicians and physicists interested in research on nonlinear phenomena, the journal''s coverage ranges from proofs of important theorems to papers presenting ideas, conjectures and numerical or physical experiments of significant physical and mathematical interest. Subject coverage: The journal publishes papers on nonlinear mathematics, mathematical physics, experimental physics, theoretical physics and other areas in the sciences where nonlinear phenomena are of fundamental importance. A more detailed indication is given by the subject interests of the Editorial Board members, which are listed in every issue of the journal. Due to the broad scope of Nonlinearity, and in order to make all papers published in the journal accessible to its wide readership, authors are required to provide sufficient introductory material in their paper. This material should contain enough detail and background information to place their research into context and to make it understandable to scientists working on nonlinear phenomena. Nonlinearity is a journal of the Institute of Physics and the London Mathematical Society.