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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the problem of linearization
for state-dependent delay differential equations


Authors: Kenneth L. Cooke and Wenzhang Huang
Journal: Proc. Amer. Math. Soc. 124 (1996), 1417-1426
MSC (1991): Primary 34K20
MathSciNet review: 1340381
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Abstract: The local stability of the equilibrium for a general class of state-dependent delay equations of the form

\begin{displaymath}\dot x(t)=f\left(x_t, \int^0_{-r_0}\,d\eta(s)g(x_t(-\tau(x_t)+s))\right)\end{displaymath}

has been studied under natural and minimal hypotheses. In particular, it has been shown that generically the behavior of the state-dependent delay $\tau$ (except the value of $\tau)$ near an equilibrium has no effect on the stability, and that the local linearization method can be applied by treating the delay $\tau$ as a constant value at the equilibrium.


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Additional Information

Kenneth L. Cooke
Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711
Email: kcooke@pomona.edu

Wenzhang Huang
Affiliation: Department of Mathematics, University of Alabama in Huntsville, Huntsville, Alabama 35899
Email: huang@math.uah.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03437-5
PII: S 0002-9939(96)03437-5
Received by editor(s): December 3, 1993
Additional Notes: The first author’s research was supported in part by NSF grant DMS 9208818
The second author’s research was supported in part by NSF grant DEB 925370 to Carlos Castillo-Chavez and by the U.S. Army Research Office through the Mathematical Science Institute of Cornell University
Communicated by: Hal L. Smith
Article copyright: © Copyright 1996 American Mathematical Society