Trajectories of a DAE near a pseudo-equilibrium

Beardmore, R. E.; Laister, Robert; Peplow, A.

Abstract

We consider a class of differential-algebraic equations (DAEs) defined by analytic nonlinearities and study its singular solutions. The main assumption used is that the linearization of the DAE represents a Kronecker index-2 matrix pencil and that the constraint manifold has a quadratic fold along its singularity. From these assumptions we obtain a normal form for the DAE where the presence of the singularity and its effects on the dynamics of the problem are made explicit in the form of a quasi-linear differential equation. Subsequently, two distinct types of singular points are identified through which there pass exactly two analytic solutions: pseudo-nodes and pseudo-saddles. We also demonstrate that a singular point called a pseudo-node supports an uncountable infinity of solutions which are not analytic in general. Moreover, akin to known results in the literature for DAEs with singular equilibria, a degenerate singularity is found through which there passes one analytic solution such that the singular point in question is contained within a quasi-invariant manifold of solutions. We call this type of singularity a pseudo-centre and it provides not only a manifold of solutions which intersects the singularity, but also a local flow on that manifold which solves the DAE.

• Journal Article Type: Article
• Publication Date: Jan 1, 2004
• Journal: Nonlinearity
• Print ISSN: 0951-7715
• Publisher: London Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 17
• Issue: 1
• Pages: 253-279
• DOI: https://doi.org/10.1088/0951-7715/17/1/015
• Keywords: differential-algebraic equations, invariant manifold, pseudo equilibrium, matrix pencil
• Public URL: https://uwe-repository.worktribe.com/output/1067114
• Publisher URL: http://dx.doi.org/10.1088/0951-7715/17/1/015