Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Walter 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, 10.1137/0152048
  • [2] Wolfgang 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
  • [3] Jacques 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
  • [4] S. 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, 10.1016/0040-5809(84)90011-X
  • [5] Kenneth 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
  • [6] Kenneth 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
  • [7] J. 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, 10.1016/0362-546X(84)90121-4
  • [8] Jack Hale, Theory of functional differential equations, 2nd ed., Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences, Vol. 3. MR 0508721
  • [9] F. 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
  • [10] Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous state-dependent delay equations, IMA Preprint Series #754, Minneapolis, 1990.
  • [11] J. 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
  • [12] H. L. Smith, Threshold delay differential equations are equivalent to FDE's, preprint.
  • [13] James C. Beidleman and Howard Smith, On Frattini-like subgroups, Glasgow Math. J. 35 (1993), no. 1, 95–98. MR 1199942, 10.1017/S0017089500009605

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34K20

Retrieve articles in all journals with MSC (1991): 34K20

Additional Information

Kenneth L. Cooke
Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711

Wenzhang Huang
Affiliation: Department of Mathematics, University of Alabama in Huntsville, Huntsville, Alabama 35899

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