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Education
SIAM Review ( IF 10.2 ) Pub Date : 2023-08-08 , DOI: 10.1137/23n975752 Hélène Frankowska
SIAM Review ( IF 10.2 ) Pub Date : 2023-08-08 , DOI: 10.1137/23n975752 Hélène Frankowska
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.
中文翻译：
教育
《SIAM 评论》，第 65 卷，第 3 期，第 867-867 页，2023 年 8 月。
在本期中，教育部分提出了两项贡献。Aminur Rahman 和 D. Blackmore 所著的“Peixoto 结构稳定性定理的一维版本：基于微积分的证明”在一维环境中提出了 Peixoto 结构稳定性和密度定理的新颖证明，该定理是基础性的在动力系统理论中。在此框架中，结构稳定性定理表明，$\mathbb{S} ^1$ 上的 $C^1$ 动力系统 $\dot x =f(x)$ 是结构稳定的，当且仅当它具有有限多个平衡点，所有这些都是双曲的。在上面的 $\mathbb{S} ^1$ 表示 $\mathbb{R}^2$ 中的单位圆，如果 $f'(x_\star) \neq 0$ ，点 $x_\star$ 称为双曲。Peixto 密度结果表明，$\mathbb{S}^1$ 上的所有 $C^1$ 结构稳定系统的集合在 $\mathbb{S 上的所有 $C^1$ 动力系统的空间中是开且稠密的} ^1$ 赋予$C^1$ 范数。原始的 Peixoto 定理更为复杂，并且对于任何光滑的闭合曲面都有效。然而，它的证明很长，并且无法使用高年级本科生可以使用的工具来获得，这与本科生可以遵循的所提出的一维证明形成鲜明对比。这并不意味着证明本身是基本的。预备知识回顾成功完成任务所需的所有基本定义。风格严谨，自成一体。文章还提供了一些历史评论，使阅读变得生动活泼，鼓励进一步学习。第二篇论文，“儿童秋千的分段平滑模型”由 Brigid Murphy 和 Paul Glendinning 介绍。它涉及一个儿童模型，无论是坐着还是站着，在操场上荡秋千。在这篇来自一位作者的硕士学位论文的文章中，这些模型使用拉格朗日力学进行了分析，并且可以作为分段平滑系统在建模中出现的不同方式的介绍。作者描述了挥杆者的控制策略，特别是挥杆是否有可能在其枢轴上完成一个完整的转动。在讨论策略时，自然会出现分段平滑项，并且对这个未来进行了详细分析。事实上，挥杆者身体的瞬时重新定位会导致挥杆姿势的跳跃。使用标准软件包进行数值模拟。这些研究适用于与经典力学课程相关的本科项目。在更高层次上，项目可以包括进一步完善现有方法和/或使用可用的专用软件包获得更准确的数值解决方案。最后一节还讨论了在这种情况下研究起来很有趣的各种相关数学问题，并提到了涉及使用分段平滑项描述的跳跃的其他模型。项目可能包括进一步完善现有方法和/或使用可用的专用软件包获得更准确的数值解决方案。最后一节还讨论了在这种情况下研究起来很有趣的各种相关数学问题，并提到了涉及使用分段平滑项描述的跳跃的其他模型。项目可能包括进一步完善现有方法和/或使用可用的专用软件包获得更准确的数值解决方案。最后一节还讨论了在这种情况下研究起来很有趣的各种相关数学问题，并提到了涉及使用分段平滑项描述的跳跃的其他模型。
更新日期：2023-08-08
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.
中文翻译：
教育
《SIAM 评论》，第 65 卷，第 3 期，第 867-867 页，2023 年 8 月。
在本期中，教育部分提出了两项贡献。Aminur Rahman 和 D. Blackmore 所著的“Peixoto 结构稳定性定理的一维版本：基于微积分的证明”在一维环境中提出了 Peixoto 结构稳定性和密度定理的新颖证明，该定理是基础性的在动力系统理论中。在此框架中，结构稳定性定理表明，$\mathbb{S} ^1$ 上的 $C^1$ 动力系统 $\dot x =f(x)$ 是结构稳定的，当且仅当它具有有限多个平衡点，所有这些都是双曲的。在上面的 $\mathbb{S} ^1$ 表示 $\mathbb{R}^2$ 中的单位圆，如果 $f'(x_\star) \neq 0$ ，点 $x_\star$ 称为双曲。Peixto 密度结果表明，$\mathbb{S}^1$ 上的所有 $C^1$ 结构稳定系统的集合在 $\mathbb{S 上的所有 $C^1$ 动力系统的空间中是开且稠密的} ^1$ 赋予$C^1$ 范数。原始的 Peixoto 定理更为复杂，并且对于任何光滑的闭合曲面都有效。然而，它的证明很长，并且无法使用高年级本科生可以使用的工具来获得，这与本科生可以遵循的所提出的一维证明形成鲜明对比。这并不意味着证明本身是基本的。预备知识回顾成功完成任务所需的所有基本定义。风格严谨，自成一体。文章还提供了一些历史评论，使阅读变得生动活泼，鼓励进一步学习。第二篇论文，“儿童秋千的分段平滑模型”由 Brigid Murphy 和 Paul Glendinning 介绍。它涉及一个儿童模型，无论是坐着还是站着，在操场上荡秋千。在这篇来自一位作者的硕士学位论文的文章中，这些模型使用拉格朗日力学进行了分析，并且可以作为分段平滑系统在建模中出现的不同方式的介绍。作者描述了挥杆者的控制策略，特别是挥杆是否有可能在其枢轴上完成一个完整的转动。在讨论策略时，自然会出现分段平滑项，并且对这个未来进行了详细分析。事实上，挥杆者身体的瞬时重新定位会导致挥杆姿势的跳跃。使用标准软件包进行数值模拟。这些研究适用于与经典力学课程相关的本科项目。在更高层次上，项目可以包括进一步完善现有方法和/或使用可用的专用软件包获得更准确的数值解决方案。最后一节还讨论了在这种情况下研究起来很有趣的各种相关数学问题，并提到了涉及使用分段平滑项描述的跳跃的其他模型。项目可能包括进一步完善现有方法和/或使用可用的专用软件包获得更准确的数值解决方案。最后一节还讨论了在这种情况下研究起来很有趣的各种相关数学问题，并提到了涉及使用分段平滑项描述的跳跃的其他模型。项目可能包括进一步完善现有方法和/或使用可用的专用软件包获得更准确的数值解决方案。最后一节还讨论了在这种情况下研究起来很有趣的各种相关数学问题，并提到了涉及使用分段平滑项描述的跳跃的其他模型。
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