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Phase portraits, Lyapunov functions, and projective geometry. Lessons learned from a differential equations class and its aftermath. (English) Zbl 1465.34041

Math. Semesterber. 68, No. 1, 143-161 (2021); erratum ibid. 68, No. 1, 163 (2021).
Summary: 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.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
97D40 Mathematics teaching methods and classroom techniques
97E50 Reasoning and proving in the mathematics classroom
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