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

Werner M Seiler,Matthias Seiß,Thomas Sturm

doi:10.1007/s11786-020-00485-x

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.

Highlights

Implicit differential equations, i. e. equations which are not solved for a derivative of highest order, appear in many applications
This is on the one hand more restrictive, as only polynomial equations and inequalities are allowed. It is more general, as a semialgebraic set may have singularities in the sense of algebraic geometry. We will call such points algebraic singularities of the semialgebraic differential equation J to distinguish them from the geometric singularities on which we focus in this work
For the transition from differential algebra to jet geometry, we introduce for any finite order ∈ N the finitedimensional subrings D = D ∩ R[t, u, . . . , u( )]

Summary

Introduction

I. e. equations which are not solved for a derivative of highest order, appear in many applications. We will make stronger use of the fact that the detection of singularities represents essentially a linear problem This will allow us to avoid some redundant case distinctions that are unavoidable in the approach of [33], as they must appear in any algebraic Thomas decomposition, they are irrelevant for the detection of singularities. These examples are fairly small, it becomes evident how our logic based approach avoids some unnecessary case distinctions made by the algebraic Thomas decomposition

Geometric Singularities of Implicit Ordinary Differential Equations

Parametric Gaussian Elimination

Detecting Geometric Singularities with Logic

Computational Experiments

Conclusions