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

Abstract

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.

Singularities of Algebraic Differential Equations

“…This process represents a suitable alternative for the effective detection of real singularities and as a by-product avoids to some extent the above mentioned problem that the algebraic Thomas decomposition leads to unnecessary case distinctions because of the geometry of the embedding of the differential equation. As demonstrated in [SSS20], an analysis of Example 7.2 leads now to no redundant cases.…”

“…From an application point of view, it is of great interest to have a similar theory as developed in this work for real differential equations. A first step in this direction can be found in [SSS20] for ordinary differential equations. There the algebraic Thomas decomposition is replaced by a parametric Gaussian algorithm followed by a quantifier elimination.…”