n 维片断多项式矢量场的庞加莱压缩:理论与应用

IF 0.6 4区 数学 Q3 MATHEMATICS
Shimin Li , Jaume Llibre , Qian Tong
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引用次数: 0

摘要

庞加莱致密化对于研究无穷邻域矢量场的动力学非常重要,这也是天体力学、天体物理学、天文学和某些化学分支对粒子逸出无穷的主要关注点。从那时起,Poincaré 压缩被扩展到各种情况,如:n 维多项式矢量场、哈密顿矢量场、准均质矢量场、有理矢量场等。近年来,描述具有不连续性的情况(如切换、决策、冲击等)的片状光滑矢量场越来越受到关注。值得注意的是,Poincaré compactification 已成功地扩展到 2 维和 3 维的片断多项式向量场,也有关于 n 维 Lipschitz 连续向量场的研究。本文的主要目标是将普恩卡雷致密化扩展到 n 维的片断多项式矢量场,因为这些矢量场通常是不连续的,而这正是现有文献中缺少的一点。因此,我们可以研究 n 维片断多项式矢量场在无穷大附近的动力学。作为应用,我们研究了一类三维片断线性微分系统的全局相位肖像。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Poincaré compactification for n-dimensional piecewise polynomial vector fields: Theory and applications
Poincaré compactification is very important to investigate the dynamics of vector fields in the neighborhood of the infinity, which is the main concern on the escape of particles to infinity in celestial mechanics, astrophysics, astronomy and some branches of chemistry. Since then Poincaré compactification has been extended into various cases, such as: n-dimensional polynomial vector fields, Hamiltonian vector fields, quasi-homogeneous vector fields, rational vector fields, etc.
In recent years, the piecewise smooth vector fields describing situations with discontinuities such as switching, decisions, impacts etc., have been attracted more and more attention. It is worth to notice that Poincaré compactification has been extended successfully to piecewise polynomial vector fields in 2-dimensional and 3-dimensional cases, and there are also works on n-dimensional Lipschitz continuous vector fields. The main goal of present paper is to extend the Poincaré compactification to n-dimensional piecewise polynomial vector fields which are usually discontinuous, this is a missing point in the existent literature. Thus we can investigate the dynamics near the infinity of n-dimensional piecewise polynomial vector fields. As an application we study the global phase portraits for a class of 3-dimensional piecewise linear differential systems.
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来源期刊
CiteScore
1.20
自引率
33.30%
发文量
251
审稿时长
6 months
期刊介绍: Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.
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