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.
See this paper.
This paper concerns quasi-linear implicit differential equations of form 0=A1(x)x'- g1(x), 0=g2(x), where A1:U->L(Rn,Rn-m) and g1:U->Rn-m are of class C1, g2:U->Rm is of class C2, U is an open subset of Rn, n,m are natural numbers, m is less than n. In particular, the above differential equation is considered about impasse points x0 in U, i.e., points x0 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 x1r x1' = sigma, x2' = 0, ..., xn-m' = 0, xn-m+1=0, ..., xn=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:
 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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