Education

Abstract

SIAM Review, Volume 65, Issue 3, Page 867-867, August 2023.
In this issue, the Education section presents two contributions. “The One-Dimensional Version of Peixoto's Structural Stability Theorem: A Calculus-Based Proof,” by Aminur Rahman and D. Blackmore, proposes, in the one-dimensional setting, a novel proof of Peixoto's structural stability and density theorem, which is fundamental in dynamical systems theory. In this framework the structural stability theorem says that a $C^1$ dynamical system $\dot x =f(x)$ on $\mathbb{S} ^1$ is structurally stable if and only if it has finitely many equilibrium points, all of which are hyperbolic. In the above $\mathbb{S} ^1$ denotes the unit circle in $\mathbb{R}^2$ and a point $x_\star$ is called hyperbolic if $f'(x_\star) \neq 0$. The Peixto density result says that the set of all $C^1$ structurally stable systems on $\mathbb{S}^1$ is open and dense in the space of all $C^1$ dynamical systems on $\mathbb{S} ^1$ endowed with the $C^1$ norm. The original Peixoto's theorem is more complex and is valid for any smooth closed surface. Its proof, however, is long and not accessible using the tools available to advanced undergraduates, in contrast with the proposed one-dimensional proof, which an undergraduate could follow. This does not mean that the proof itself is elementary. Preliminaries recall all the basic definitions that are needed to successfully conduct the task. The style is rigorous and self-contained. The article also provides some historical comments, making the reading lively and encouraging further learning. The second paper, “Piecewise Smooth Models of Pumping a Child's Swing,” is presented by Brigid Murphy and Paul Glendinning. It concerns models of a child, in either a seated or standing position, swinging on a playground swing. In the article, which arose from the MSc dissertation by one of the authors, these models are analyzed using Lagrangian mechanics and may serve as an introduction to the different ways in which piecewise smooth systems do arise in modeling. The authors describe control strategies of swingers, and, in particular, whether it is possible for the swing to go through a full turn over its pivot. Piecewise smooth terms do naturally appear while discussing the strategies, and this future is analyzed in detail. Indeed the instantaneous reposition of the body of the swinger leads to a jump in the configuration of the swing. Numerical simulations are performed with a standard software package. These investigations would be suitable for undergraduate projects related to classical mechanics courses. At a higher degree level, projects could include further refinement of the existing methods and/or getting more accurate numerical solutions using available specialized software packages. The final section also discusses various related mathematical questions that would be interesting to investigate in this context and mentions other models involving jumps described using piecewise smooth terms.