A Logic Based Approach to Finding Real Singularities of Implicit Ordinary Differential Equations

• Published in: Mathematics in computer science Vol. 15; no. 2; pp. 333 - 352
• Main Authors: Seiler, Werner M., Seiß, Matthias, Sturm, Thomas
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2021; Springer Nature B.V; Springer
• Subjects: Algebraic Geometry; Biochemistry, Molecular Biology; Computer Science; Differential equations; Differential geometry; Dynamical Systems; Gaussian elimination; Life Sciences; Logic in Computer Science; Mathematical analysis; Mathematics; Mathematics and Statistics; Molecular Networks; Ordinary differential equations; Real numbers; Singularities; Symbolic Computation; Logic computation; 68W30; Primary 34A09; 34A26; Implicit differential equations; Geometric singularities; 37C10; Secondary 34-04; Vessiot distribution; 34C40; Real algebraic computations; 34C08; Logic computation Mathematics Subject Classification Primary 34A09
• ISSN: 1661-8270; 1661-8289
• DOI: 10.1007/s11786-020-00485-x

We discuss the effective computation of geometric singularities of implicit ordinary differential equations over the real numbers using methods from logic. Via the Vessiot theory of differential equations, geometric singularities can be characterised as points where the behaviour of a certain linear system of equations changes. These points can be discovered using a specifically adapted parametric generalisation of Gaussian elimination combined with heuristic simplification techniques and real quantifier elimination methods. We demonstrate the relevance and applicability of our approach with computational experiments using a prototypical implementation in Reduce .