The One-dimensional Version of Peixoto's Structural Stability Theorem: A Calculus-based Proof

TL;DR

This paper presents a simplified, calculus-based proof of the one-dimensional analogs of Peixoto's structural stability theorems, making these complex results accessible to upper-level undergraduates and highlighting their historical significance.

Contribution

It provides an intuitive, accessible proof of one-dimensional Peixoto theorems, bridging advanced dynamical systems theory and undergraduate education, with historical context.

Findings

Simplified proof of one-dimensional Peixoto's theorems

Accessible explanation suitable for undergraduates

Highlights historical significance of the theorems

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 analogs 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\'ilia Chaves Peixoto and Mauricio Matos Peixoto's births, February 24th and April 15th 1921, respectively.