Peixoto's Structural Stability Theorem: The One-dimensional Version

TL;DR

This paper presents an accessible formulation and proof of the one-dimensional version of Peixoto's structural stability theorems, making these fundamental results in dynamical systems more understandable for upper-level undergraduates.

Contribution

It simplifies and clarifies the one-dimensional analogs of Peixoto's theorems, making them accessible to students with basic mathematical background.

Findings

Provides an intuitive proof of the one-dimensional Peixoto's theorems

Makes structural stability concepts more accessible to undergraduates

Facilitates further study in dynamical systems for students

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 note/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.