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The One-Dimensional Version of Peixoto's Structural Stability Theorem: A Calculus-Based Proof
SIAM Review ( IF 10.2 ) Pub Date : 2023-08-08 , DOI: 10.1137/21m1426572 Aminur Rahman , D. Blackmore
SIAM Review ( IF 10.2 ) Pub Date : 2023-08-08 , DOI: 10.1137/21m1426572 Aminur Rahman , D. Blackmore
SIAM Review, Volume 65, Issue 3, Page 869-886, August 2023.
Peixoto's structural stability and density theorems represent milestones in the modern theory of dynamical systems and their applications. Despite the importance of these theorems, they are often treated rather superficially, if at all, in upper level undergraduate courses on dynamical systems or differential equations. This is mainly because of the depth and length of the proofs. In this module, we formulate and prove the one-dimensional analogues of Peixoto's theorems in an intuitive and fairly simple way using only concepts and results that for the most part should be familiar to upper level undergraduate students in the mathematical sciences or related fields. The intention is to provide students who may be interested in further study in dynamical systems with an accessible one-dimensional treatment of structural stability theory that should help make Peixoto's theorems and their more recent generalizations easier to appreciate and understand. Further, we believe it is important and interesting for students to know the historical context of these discoveries since the mathematics was not done in isolation. The historical context is perhaps even more appropriate as it is the 100th anniversary of Marília Chaves Peixoto's and Maurício Matos Peixoto's births, February 24th and April 15th, 1921, respectively.
中文翻译：
Peixoto 结构稳定性定理的一维版本：基于微积分的证明
《SIAM 评论》，第 65 卷，第 3 期，第 869-886 页，2023 年 8 月。
佩肖托的结构稳定性和密度定理代表了现代动力系统理论及其应用的里程碑。尽管这些定理很重要，但在动力系统或微分方程的高年级本科课程中，如果有的话，它们常常被相当肤浅地对待。这主要是因为样张的深度和长度。在本模块中，我们仅使用数学科学或相关领域的高年级本科生应该熟悉的概念和结果，以直观且相当简单的方式制定和证明 Peixoto 定理的一维类似物。目的是为有兴趣进一步研究动力系统的学生提供结构稳定性理论的一种易于理解的一维处理方式，这应该有助于使 Peixoto 定理及其最近的概括更容易理解和理解。此外，我们认为，对于学生来说，了解这些发现的历史背景是重要且有趣的，因为数学并不是孤立进行的。历史背景也许更合适，因为今天是 Marília Chaves Peixoto 和 Maurício Matos Peixoto 诞辰 100 周年，分别是 1921 年 2 月 24 日和 4 月 15 日。我们相信，对于学生来说，了解这些发现的历史背景是重要且有趣的，因为数学并不是孤立进行的。历史背景也许更合适，因为今天是 Marília Chaves Peixoto 和 Maurício Matos Peixoto 诞辰 100 周年，分别是 1921 年 2 月 24 日和 4 月 15 日。我们相信，对于学生来说，了解这些发现的历史背景是重要且有趣的，因为数学并不是孤立进行的。历史背景也许更合适，因为今天是 Marília Chaves Peixoto 和 Maurício Matos Peixoto 诞辰 100 周年，分别是 1921 年 2 月 24 日和 4 月 15 日。
更新日期：2023-08-08
Peixoto's structural stability and density theorems represent milestones in the modern theory of dynamical systems and their applications. Despite the importance of these theorems, they are often treated rather superficially, if at all, in upper level undergraduate courses on dynamical systems or differential equations. This is mainly because of the depth and length of the proofs. In this module, we formulate and prove the one-dimensional analogues of Peixoto's theorems in an intuitive and fairly simple way using only concepts and results that for the most part should be familiar to upper level undergraduate students in the mathematical sciences or related fields. The intention is to provide students who may be interested in further study in dynamical systems with an accessible one-dimensional treatment of structural stability theory that should help make Peixoto's theorems and their more recent generalizations easier to appreciate and understand. Further, we believe it is important and interesting for students to know the historical context of these discoveries since the mathematics was not done in isolation. The historical context is perhaps even more appropriate as it is the 100th anniversary of Marília Chaves Peixoto's and Maurício Matos Peixoto's births, February 24th and April 15th, 1921, respectively.
中文翻译：
Peixoto 结构稳定性定理的一维版本：基于微积分的证明
《SIAM 评论》，第 65 卷，第 3 期，第 869-886 页，2023 年 8 月。
佩肖托的结构稳定性和密度定理代表了现代动力系统理论及其应用的里程碑。尽管这些定理很重要，但在动力系统或微分方程的高年级本科课程中，如果有的话，它们常常被相当肤浅地对待。这主要是因为样张的深度和长度。在本模块中，我们仅使用数学科学或相关领域的高年级本科生应该熟悉的概念和结果，以直观且相当简单的方式制定和证明 Peixoto 定理的一维类似物。目的是为有兴趣进一步研究动力系统的学生提供结构稳定性理论的一种易于理解的一维处理方式，这应该有助于使 Peixoto 定理及其最近的概括更容易理解和理解。此外，我们认为，对于学生来说，了解这些发现的历史背景是重要且有趣的，因为数学并不是孤立进行的。历史背景也许更合适，因为今天是 Marília Chaves Peixoto 和 Maurício Matos Peixoto 诞辰 100 周年，分别是 1921 年 2 月 24 日和 4 月 15 日。我们相信，对于学生来说，了解这些发现的历史背景是重要且有趣的，因为数学并不是孤立进行的。历史背景也许更合适，因为今天是 Marília Chaves Peixoto 和 Maurício Matos Peixoto 诞辰 100 周年，分别是 1921 年 2 月 24 日和 4 月 15 日。我们相信，对于学生来说，了解这些发现的历史背景是重要且有趣的，因为数学并不是孤立进行的。历史背景也许更合适，因为今天是 Marília Chaves Peixoto 和 Maurício Matos Peixoto 诞辰 100 周年，分别是 1921 年 2 月 24 日和 4 月 15 日。
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