Gunther Reißig and Holger Boche.

Singularities of implicit ordinary differential equations.

Proc. 1998 IEEE Int. Symp. Circ. Systems (ISCAS), Monterey, CA, May 31 - June 3, 1998, vol. 3, pp. 326-329.

**Erratum:**

See this paper.

**Abstract:**

This paper concerns quasi-linear implicit
differential equations of form
0=A_{1}(x)x'- g_{1}(x), 0=g_{2}(x),
where A_{1}:U->L(R^{n},R^{n-m}) and g_{1}:U->R^{n-m} are of class C^{1},
g_{2}:U->R^{m} is of class C^{2}, U is an open subset of R^{n},
n,m are natural numbers, m is less than n.
In particular, the above differential equation is considered about
impasse points x_{0} in U, i.e., points x_{0} beyond
which solutions are not continuable.
Under appropriate assumptions, it is shown that
there is a diffeomorphism
that transforms solutions of the above implicit differential equation
near such points into solutions of the normal form
x_{1}^{r} x_{1}' = sigma,
x_{2}' = 0, ..., x_{n-m}' = 0,
x_{n-m+1}=0, ..., x_{n}=0
near 0, and vice versa, where sigma=+/-1=const.
In particular, standard impasse points in the sense of Rabier and Rheinboldt
lead to the above normal form with r=1.
A practical example for r=2 is also given.

**BibTeX entry:**

@inproceedings{ReissigBoche98b,
author = {Gunther Rei{\ss}ig and Holger Boche},
title = {Singularities of implicit ordinary differential equations},
booktitle = {Proc. 1998 IEEE Int. Symp. on Circuits and Systems (ISCAS), Monterey, CA, } # may # { 31 -- } # jun # { 3},
year = 1998,
volume = 3,
pages = {326-329},
doi = {10.1109/ISCAS.1998.704016}
}

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