Current and past semesters
- Galois Theories of Differential Equations and Differential Algebra.
- Analysis of Singularities of General Systems of Differential Equations Using Techniques from Differential Algebra, Algebraic Geometry and Differential Topology.
Preprints and Papers
- Normal Forms in Differential Galois Theory for the Classical Groups with Daniel Robertz.
Let G be a classical group of dimension d and let a=(a1,…,ad) be differential indeterminates over a differential field F of characteristic zero with algebraically closed field of constants C. Further let A(a) be a generic element in the Lie algebra 𝔤(F〈a〉) of G obtained from parametrizing a basis of 𝔤 with the indeterminates a. It is known (cf. work of Juan) that the differential Galois group of ∂(y)=A(a)y over F〈a〉 is G(C). In this paper we construct a differential field extension L of F〈a〉 such that the field of constants of L is C, the differential Galois group of ∂(y)=A(a)y over L is still the full group G(C) and A(a) is gauge equivalent over L to a matrix in normal form which we introduced in work of Seiss. We also consider specializations of the coefficients of A(a).
- On General Extension Fields for the Classical Groups in Differential Galois Theory.
Let G be one of the classical groups of Lie rank l. We make a similar construction of a general extension field in differential Galois theory
for G as E. Noether did in classical Galois theory for finite groups. More precisely, we build a differential field E of differential transcendence degree l over the constants
on which the group G acts and show that it is a Picard-Vessiot extension of the field of invariants EG. The field EG is differentially generated by l differential polynomials which are differentially algebraically independent over the constants. They are the coefficients of the defining equation of the extension. Finally we show that our construction satisfies generic properties for a specific kind of G-primitive Picard-Vessiot extensions.
- Algebraic and Geometric Analysis of Singularities of Implicit Differential Equations with Werner M. Seiler.
Computer Algebra in Scientific Computing - CASC 2020, F. Boulier, M. England, T. Sadykov, E.V. Vorozhtsov (eds.),
LNCS 12291, Springer-Verlag, Cham 2020, pp. 14-41.
We review our recent works on singularities of implicit ordinary or partial differential equations. This includes firstly the development of a general framework combining algebraic and geometric methods for dealing with general systems of ordinary or partial differential equations and for defining the type of singularities considered here. We also present an algorithm for detecting all singularities of an algebraic differential equation over the complex numbers. We then discuss the adaptions required for the analysis over the real numbers. We further outline for a class of singular initial value problems for a second-order ordinary differential equation how geometric methods allow us to determine the local solution behaviour in the neighbourhood of a singularity including the regularity of the solution. Finally, we show for some simple cases of algebraic singularities how there such an analysis can be performed.
- Singularities of Algebraic Differential Equations with Markus Lange-Hegermann, Daniel Robertz and Werner M. Seiler.
Advances in Applied Mathematics, Volume 131, October 2021, 102266,
We combine algebraic and geometric approaches to general systems of algebraic ordinary or partial differential equations to provide a unified framework for the definition and detection of singularities of a given system at a fixed order. Our three main results are firstly a proof that even in the case of partial differential equations regular points are generic. Secondly, we present an algorithm for the effective detection of all singularities at a given order or, more precisely, for the determination of a regularity decomposition. Finally, we give a rigorous definition of a regular differential equation, a notion that is ubiquitous in the geometric theory of differential equations, and show that our algorithm extracts from each prime component a regular differential equation. Our main algorithmic tools are on the one hand the algebraic resp. differential Thomas decomposition and on the other hand the Vessiot theory of differential equations.
- A Logic Based Approach to Finding Real Singularities of Implicit Ordinary Differential Equations with Thomas Sturm and Werner M. Seiler.
Mathematics in Computer Science 15, 333-352 (2021),
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.
- No Chaos in Dixon's System with Werner M. Seiler.
International Journal of Bifurcation and Chaos, Vol. 31, No. 03, 2150044 (2021),
The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behaviour, if its two parameters take their value in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon's system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into sixteen different regions in each of which the system exhibits qualitatively the same behaviour. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist which can easily create in numerical computations the impression of chaotic behaviour.
- Singular Initial Value Problems for Scalar Quasi-Linear Ordinary Differential Equations with Werner M. Seiler.
Journal of Differential Equations, Volume 281, 25 April 2021, Pages 258-288,
We discuss existence, non-uniqueness and regularity of one- and two-sided solutions of initial value problems for scalar quasi-linear ordinary differential equations where the initial condition corresponds to an impasse point of the equation. With a differential geometric approach, we reduce the problem to questions in dynamical systems theory. As an application, we discuss in detail second-order equations of the form g(x)u''=f(x,u,u') with an initial condition imposed at a simple zero of g. This generalises results by Liang and also makes them more transparent via our geometric approach.
- On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations with Elishan Braun and Werner M. Seiler.
Mathematics in Computer Science 14, 281-293 (2020), https://doi.org/10.1007/s11786-019-00423-6.
We discuss how the geometric theory of differential equations can be used for the numerical integration and visualisation of implicit ordinary differential equations, in particular around singularities of the equation. The Vessiot theory automatically transforms an implicit differential equation into a vector field distribution on a manifold and thus reduces its analysis to standard problems in dynamical systems theory like the integration of a vector field and the determination of invariant manifolds. For the visualisation of low-dimensional situations we adapt the streamlines algorithm of Jobard and Lefer to 2.5 and 3 dimensions. A concrete implementation in Matlab is discussed and some concrete examples are presented.
- Root Parametrized Differential Equations for the Classical Groups.
Let C〈t〉 be the differential field generated by l differential indeterminates t=(t1,…,tl) over an algebraically closed field C of characteristic zero. We develop a lower bound criterion for the differential Galois group G(C) of a matrix parameter differential equation ∂(y)=A(t)y over C〈t〉 and we prove that every connected linear algebraic group is the Galois group of a linear parameter differential equation over C〈t1〉. As a second application we compute explicit and nice linear parameter differential equations over C〈t〉 for the groups SLl+1(C), SP2l(C), SO2l+1(C), SO2l(C), i.e. for the classical groups of type Al, Bl, Cl, Dl, and for G2 (here l=2).
- A Root Parametrized Differential Equation for the Special Linear Group.
Let C〈t〉 be the differential field generated by l differential indeterminates t=(t1,…,tl) over an algebraically closed field C of characteristic zero. In this article we present an explicit linear parameter differential equation over C〈t〉 with differential Galois group SLl+1(C) and show that it is a generic equation in the following sense: If F is an algebraically closed differential field with constants C and E/F is a Picard-Vessiot extension with differential Galois group H(C) ⊆ SLl+1(C), then a specialization of our equation defines a Picard-Vessiot extension differentially isomorphic to E/F.