Trajectories of a DAE near a pseudo-equilibrium
Beardmore, R. E. ; Laister, R. and Peplow, A. LU
(2004)
In Nonlinearity
17(1).
p.253-279
- 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... (More)
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.
(Less)
- author
- Beardmore, R. E.
; Laister, R.
and Peplow, A.
LU
- publishing date
- 2004-01
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Nonlinearity
- volume
- 17
- issue
- 1
- pages
- 27 pages
- publisher
- IOP Publishing
- external identifiers
-
- scopus:0346671217
- ISSN
- 0951-7715
- DOI
- 10.1088/0951-7715/17/1/015
- language
- English
- LU publication?
- no
- additional info
- Copyright: Copyright 2008 Elsevier B.V., All rights reserved.
- id
- d5b9019a-401c-4d4f-8ab3-095432f032df
- date added to LUP
- 2021-03-08 15:18:30
- date last changed
- 2025-04-04 14:23:53
@article{d5b9019a-401c-4d4f-8ab3-095432f032df,
abstract = {{<p>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.</p>}},
author = {{Beardmore, R. E. and Laister, R. and Peplow, A.}},
issn = {{0951-7715}},
language = {{eng}},
number = {{1}},
pages = {{253--279}},
publisher = {{IOP Publishing}},
series = {{Nonlinearity}},
title = {{Trajectories of a DAE near a pseudo-equilibrium}},
url = {{http://dx.doi.org/10.1088/0951-7715/17/1/015}},
doi = {{10.1088/0951-7715/17/1/015}},
volume = {{17}},
year = {{2004}},
}