Quasilinear Differential-algebraic Equations Research Articles

Classification of Codimension-1 Singular Bifurcations in Low-Dimensional DAEs

The study of bifurcations of differential-algebraic equations (DAEs) is the topic of interest for many applied sciences, such as electrical engineering, robotics, etc. While some of them were investigated already, the full classification of such bifurcations has not been done yet. In this paper, we consider bifurcations of quasilinear DAEs with a singularity and provide a full list of all codimension-one bifurcations in lower-dimensional cases. Among others, it includes singularity-induced bifurcations (SIBs), which occur when an equilibrium branch intersects a singular manifold causing certain eigenvalues of the linearized problem to diverge to infinity. For these and other bifurcations, we construct the normal forms, establish the non-degeneracy conditions and give a qualitative description of the dynamics. Also, we study singular homoclinic and heteroclinic bifurcations, which were not considered before.

Regularization of Quasi-Linear Differential-Algebraic Equations

The complete virtual design of dynamical systems, e.g., mechanical systems, electrical circuits, flow problems, or whole production processes, plays a key role in our technological progress. In this context differential-algebraic equations (DAEs) are a very important tool for the analysis and simulation of such dynamical processes. It is well known that the numerical treatment of DAEs is nontrivial in general. The occurrence of hidden constraints (contained in the DAE but only obtainable by differentiating of (parts of) the DAE) impose additional consistency conditions on the initial values and provoke severe difficulties in the direct numerical integration of DAEs as for example drift, instabilities, or convergence problems. Therefore, it is necessary to regularize or remodel the model equations before a robust numerical integration is possible.In this article we present regularization methods for quasi-linear DAEs. In particular, we discuss the projected strangeness-free formulation, the minimally extended formulation, and the overdetermined formulation as regularizations. At the end we give some remarks on the numerical treatment of the presented regularizations.

Differential algebraic equations with properly stated leading terms

In this paper, we study differential algebraic equations (DAEs) of the form A(χ, t)(d(χ, t))′+ b(χ, t) = 0 with in some sense well-matched matrix functions A(χ, t) and D(χ, t) := d′ χ (χ, t) as they arise, e.g., in circuit simulation. We characterize index 1 DAEs in this context. After analyzing those index 1 equations themselves, we apply Runge-Kutta methods and BDFs, provide stability inequalities, and show convergence. The cases of the image space of D(χ, t) or the nullspace of A(χ, t) remaining constant are pointed out to be essentially favourable for the qualitative behaviour of the approximations on long intervals. Hence, when modelling with DAEs one should try for those, constant subspaces. Relations to quasilinear DAEs in standard formulation E(χ, t)χ′ + ƒ (χ, t) = 0 are considered, too.

The Hopf bifurcation theorem for quasilinear differential-algebraic equations

We prove a generalization of the Hopf bifurcation theorem for quasilinear differential equations (DAEs), i.e. equations of the form A( μ, χ) χ = G( μ, χ) where the matrix A( χ, μ) has constant but not full rank and hence the system cannot be made into an explicit ODE. The paper includes an appendix by J. Ernsthausen addressing the numerical calculation of the Hopf points in the DAE setting.

Regularizations of Differential-Algebraic Equations Revisited

The present paper deals with quasilinear differential-algebraic equations with index 2. These equations are approximated by regularization methods. Such methods lead to singularly perturbed differential-algebraic equations. Using a geometric theory of singular perturbations convergence of the solutions of the regularized problems towards that of the index 2 problem is proved. The limits of the present theory are discussed and directions of future research are proposed.

On the computation of impasse points of quasilinear differential-algebraic equations

We present computational algorithms for the calculation of impasse points and higher-order singularities in quasi-linear differential-algebraic equations. Our method combines a reduction step, transforming the DAE into a singular ODE, with an augmentation procedure inspired by numerical bifurcation theory. Singularities are characterized by the vanishing of a scalar quantity that may be monitored along any trajectory. Two numerical examples with physical relevance are given.

On the Computation of Impasse Points of Quasi-Linear Differential-Algebraic Equations

Abstract : We present computational algorithms for the calculation of impasse points and higher order singularities in quasilinear differential-algebraic equations. Our method combines a reduction step transforming the DAE into a singular ODE with an augmentation procedure inspired by numerical bifurcation theory. Singularities are characterized by the vanishing of a scalar quantity that may be monitored along any trajectory. Two numerical examples with physical relevance are given.

On Impasse Points of Quasilinear Differential-Algebraic Equations

Abstract : This paper presents a mathematical characterization of the impasse points of quasilinear DAE's A(x) = G(x), where then A(x) is a nxn matrix having constant but not full r<n in the domain of interest. We show that a reduction procedure permits to reduce the DAE to a quasilinear ODE Al(xi)xi'=G1(xi) Rr and that impasse points correspond to special singularities of the reduced ODE. Impasse points of higher index DAE's can also be defined; the only difference is that the above reduction procedure takes more than one step.