The One-Dimensional Version of Peixoto’s Structural Stability Theorem: A Calculus-Based Proof

Document Type

Article

Publication Date

1-1-2023

Abstract

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 Mar\'{\i}lia Chaves Peixoto's and Maur\'{\i}cio Matos Peixoto's births, February 24th and April 15th, 1921, respectively.

Identifier

85169934221 (Scopus)

Publication Title

SIAM Review

External Full Text Location

https://doi.org/10.1137/21M1426572

ISSN

00361445

First Page

869

Last Page

886

Issue

3

Volume

65

Fund Ref

Department of Mathematical Sciences, Anchor University Lagos

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