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

Abstract

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.