On the problem of linearization
for state-dependent delay differential equations
Kenneth L. Cooke and Wenzhang Huang
Proc. Amer. Math. Soc. 124 (1996), 1417-1426
Full-text PDF Free Access
Similar Articles |
Abstract: The local stability of the equilibrium for a general class of state-dependent delay equations of the form
has been studied under natural and minimal hypotheses. In particular, it has been shown that generically the behavior of the state-dependent delay (except the value of near an equilibrium has no effect on the stability, and that the local linearization method can be applied by treating the delay as a constant value at the equilibrium.
G. Aiello, H.
I. Freedman, and J.
Wu, Analysis of a model representing stage-structured population
growth with state-dependent time delay, SIAM J. Appl. Math.
52 (1992), no. 3, 855–869. MR 1163810
Alt, Periodic solutions of some autonomous differential equations
with variable time delay, Functional differential equations and
approximation of fixed points (Proc. Summer School and Conf., Univ. Bonn,
Bonn, 1978) Lecture Notes in Math., vol. 730, Springer, Berlin,
1979, pp. 16–31. MR 547978
Bélair, Population models with state-dependent delays,
Mathematical population dynamics (New Brunswick, NJ, 1989) Lecture Notes
in Pure and Appl. Math., vol. 131, Dekker, New York, 1991,
pp. 165–176. MR 1227361
P. Blythe, R.
M. Nisbet, and W.
S. C. Gurney, The dynamics of population models with distributed
maturation periods, Theoret. Population Biol. 25
(1984), no. 3, 289–311. MR 752482
L. Cooke, Functional-differential equations: Some models and
perturbation problems, Differential Equations and Dynamical Systems
(Proc. Internat. Sympos., Mayaguez, P.R., 1965) Academic Press, New York,
1967, pp. 167–183. MR 0222409
L. Cooke and Wen
Zhang Huang, A theorem of George Seifert and an equation with
state-dependent delay, Delay and differential equations (Ames, IA,
1991) World Sci. Publ., River Edge, NJ, 1992, pp. 65–77. MR 1170144
A. Gatica and Paul
Waltman, Existence and uniqueness of solutions of a
functional-differential equation modeling thresholds, Nonlinear Anal.
8 (1984), no. 10, 1215–1222. MR 763658
Hale, Theory of functional differential equations, 2nd ed.,
Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences,
Vol. 3. MR
0508721 (58 #22904)
C. Hoppensteadt and P.
Waltman, A flow mediated control model of respiration, Some
Mathematical Questions in Biology (Proc. 13th Sympos. Math. Biol., Houston,
Tex., 1979) Lectures Math. Life Sci., vol. 12, Amer. Math. Soc.,
Providence, R.I., 1979, pp. 211–218. MR 640270
Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous state-dependent delay equations, IMA Preprint Series #754, Minneapolis, 1990.
A. J. Metz and O.
Diekmann (eds.), The dynamics of physiologically structured
populations, Lecture Notes in Biomathematics, vol. 68,
Springer-Verlag, Berlin, 1986. Papers from the colloquium held in
Amsterdam, 1983. MR 860959
H. L. Smith, Threshold delay differential equations are equivalent to FDE's, preprint.
C. Beidleman and Howard
Smith, On Frattini-like subgroups, Glasgow Math. J.
35 (1993), no. 1, 95–98. MR 1199942
- W. G. Aiello, H. I. Freedman, and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math. 52, 1992, 855--869. MR 93j:92025
- W. Alt, Periodic solutions of some autonomous differential equations with variable time delay, Lecture Notes in Mathematics, Vol. 730, Springer-Verlag, 1979. MR 80i:34118
- J. Belair, Population models with state-dependent delays, In Mathematical Population Dynamics (O. Arino, D. E. Axelrod, and M. Kimmel, Eds.), Marcel Dekker, Inc., New York-Basel-Hong Kong, 1991, 165--176.MR 94e:92022
- S. P. Blythe, R. M. Nisbet, and W. S. C. Gurney, The dynamics of population models with distributed maturation periods, Theor. Pop. Biology 25, 1984, 289--311.MR 85h:92030
- K. L. Cooke, Functional differential equations: some models and perturbation problems, In Differential Equations and Dynamical Systems (J. K. Hale and J. P. LaSalle, Eds.), Academic Press, New York, 1967. MR 36:5461
- K. L. Cooke and W. Huang, A theorem of George Seifert and an equation with state-dependent delay, In Delay and Differential Equations (A. M. Fink, R. K. Miller, and W. Kliemann, Eds.), World Scientific, Singapore, 1992, 65--77. MR 93d:34128
- J. A. Gatica and P. Waltman, Existence and uniqueness of solutions of a functional differential equation modeling thresholds, Nonlinear Analysis T.M.A. 8, 1984, 1215--1222. MR 86h:34080
- J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. MR 58:22904
- F. C. Hoppensteadt and P. Waltman, A flow mediated control model of respiration, In Some Mathematical Questions in Biology, Vol. 12, Lectures on Mathematics in the Life Sciences, 1979, 211--218. MR 82k:92026
- Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous state-dependent delay equations, IMA Preprint Series #754, Minneapolis, 1990.
- J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics 68, Springer-Verlag, 1986. MR 88b:92049
- H. L. Smith, Threshold delay differential equations are equivalent to FDE's, preprint.
- H. L. Smith, Reduction of structured population models to threshold-type delay equations and functional differential equations: a case study, Math. Biosci. 113 (1993), 1--23. MR 93k:20054
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
Retrieve articles in all journals
with MSC (1991):
Kenneth L. Cooke
Department of Mathematics, Pomona College, Claremont, California 91711
Department of Mathematics, University of Alabama in Huntsville, Huntsville, Alabama 35899
Received by editor(s):
December 3, 1993
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
Hal L. Smith
© Copyright 1996
American Mathematical Society