Phase portraits, Lyapunov functions, and projective geometry
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Phase portraits, Lyapunov functions, and projective geometry Lessons learned from a differential equations class and its aftermath Lilija Naiwert · Karlheinz Spindler
Received: 16 December 2019 / Accepted: 13 September 2020 © The Author(s) 2020
Abstract We discuss two problems which grew out of an introductory differential equations class but were solved only later, each after having been put into a different context. First, how do you find a rather complicated Lyapunov function with your bare hands, without using a fully developed theory (while reconstructing the steps leading up to such a theory)? Second, how can you obtain a global picture of the phase-portrait of a dynamical system (thereby invoking ideas from projective geometry)? Since classroom experiences played an important part in the making of this paper, didactical aspects will also be discussed. Keywords Ordinary differential equations · Lyapunov functions · Projective phase-portraits · Mathematics education 2010 Mathematical Subject Classification: 34A26 · 34D20 · 37C75 · 51N15 · 97D40 · 97E50
1 Introduction In this paper we reap some late fruit from seeds sown about three years ago, in a class on “Ordinary Differential Equations and Dynamical Systems” which was taught by the second author and attended by the first author as a third-semester student. This class had an extent of 10 blocks per week of 45 minutes each (usually 6 for lectures and 4 for lab sessions) and covered the following topics: elementary types of scalar L. Naiwert · K. Spindler () Hochschule RheinMain, Wiesbaden, Germany E-Mail: [email protected] L. Naiwert E-Mail: [email protected]
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L. Naiwert, K. Spindler
differential equations, general formulation of systems of ordinary differential equations, existence and uniqueness theorems, maximal solution intervals, dependence of solutions on initial data and parameters, variational equations, systems of linear differential equations, state transition operator, qualitative theory: phase portraits, equilibria, isoclines, elementary stability theory, Lyapunov functions. (Essentially, chapters 117–124 and 134 from [21] were covered.) There were three written tests during the semester, focusing on solution techniques and computational skills, and an oral exam at the end of the semester, focusing on conceptual understanding. In the class, which was attended by about 25 students, a rather exciting and communicative atmosphere developed, with lots of questions, comments and ideas being brought up by the students. The two questions discussed in this paper came up in this class, but were settled only later, each after having been put into a different context. To convey a flavor of the way the class was taught, we present the topics in the style of the course, using an example-oriented approach, favoring explicit calculations before introducing more abstract points of view and using pictures and visualizations whenever possible. At the end of the paper, we comment on the classroom experiences made with this approach and
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