  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.
