Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T04:28:01.824Z Has data issue: false hasContentIssue false

Variation of constants for hybrid systems of functional differential equations

Published online by Cambridge University Press:  14 November 2011

Jack K. Hale
Affiliation:
Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, Georgia 30332-0190, U.S.A.
Wenzhang Huang
Affiliation:
Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, Georgia 30332-0190, U.S.A.

Abstract

The objective is to derive a variation of constants formula for systems of functional differential equations (or delay differential equations) coupled with functional equations (or difference equations). The difficulties arise because of the constraints imposed by the functional equations.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Abolinia, V. E. and Mishkis, A. D.. Mixed problems for quasi-linear hyperbolic systems in the plane. Mat. Sbornik 50:92 (1960), 423442.Google Scholar
2Brayton, R. K.. Small signal stability criterion for electrical networks containing lossless transmission lines. IBM J. Res. Devel. 12 (1968), 431440.CrossRefGoogle Scholar
3Brayton, R. K.. Bifurcation of periodic solutions in a nonlinear difference differential equation of neutral type. Quart. Appl. Math. 24 (1966), 215224.CrossRefGoogle Scholar
4Brayton, R. K. and Miranker, W. L.. A stability theory for nonlinear mixed initial boundary value problems. Arch. Rational Mech. Anal. 17 (1964), 358376.CrossRefGoogle Scholar
5Chow, S.-N., Diekmann, O. and Mallet-Paret, J.. Stability, multiplicity and glogal continuation of symmetric periodic solutions of a nonlinear Volterra integral equation. Japan J. Appl. Math. 2 (1985), 443469.CrossRefGoogle Scholar
6Chow, S.-N. and Huang, W.. Singular perturbation problems for a system of differential difference equations. J. Differential Equations (to appear).Google Scholar
7Cooke, K. L. and Krumme, D. W.. Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations. J. Math. Anal. Appl. 24 (1968), 372387.CrossRefGoogle Scholar
8Hale, J. K.. Theory of Functional Differential Equations (New York: Springer, 1977).CrossRefGoogle Scholar
9Hale, J. K. and Lunel, S.. Functional Differential Equations (Berlin: Springer, 1993).Google Scholar
10Hale, J. K. and Meyer, K. R.. A class of functional differential equations of neutral type. Mem. Amer. Math. Soc. 76 (1967), 165.Google Scholar
11Ikeda, K.. Multiple-values stationary state and its instability of the transmitted light by a ring cavity system. Opt. Commun. 30 (1979), 257261.CrossRefGoogle Scholar
12Ikeda, K., Daido, H. and Akimoto, O.. Optical turbulence: Chaotic behavior of transmitted light from a ring cavity. Phys. Rev. Lett. 45 (1980), 709712.CrossRefGoogle Scholar
13Kobyakov, I. I.. Conditions for negativity of the Green's function of a two-point boundary value problem with deviating arguments. Differentsialnye Uravneniya 8 (1972), 443452.Google Scholar
14Martinez-Amores, P.. Estabilidad y solutiones periodicas de un problema descrito por ecuaciones diferenciales funcionales de tipo neutro (Ph.D. Thesis, Departamento de Teoria de Functiones, Universidad de Granada, 1976).Google Scholar
15Martinez-Amores, P.. Periodic solutions of coupled systems of differential and difference equations. Ann. Math. Pura Appl. (4) 12 (1979), 171186.CrossRefGoogle Scholar
16Nagumo, J. and Shimura, M.. Self-oscillation in a transmission line with a tunnel diode. Proc. IRE 49(1961), 12811291.CrossRefGoogle Scholar
17Razvan, V.. Stabilitatea Absoluta a Sistemelor Automatě cu Intitziere (Bucharest: Ed. Acad. Rep. Soc. Romania, 1975).Google Scholar
18Razvan, V.. Absolute stability of a class of control processes described by functional differential equations of neutral type. In Equations Differentielles et Fonctionelles Nonlineaires (Paris: Hermann, 1973).Google Scholar
19Shimura, M.. Analysis of some nonlinear phenomena in a transmission line. IEEE Trans. Circuit Theory 14 (1967), 6068.CrossRefGoogle Scholar
20Solodovnikov, V. V.. Techniceskaya Kibernatika, Vol. 2, Chap. XI (Moscow: Masinostroenie, 1967).Google Scholar
21Wu, J. and Xia, H.. Self-sustained oscillations in a ring array of coupled lossless transmission lines (Preprint).Google Scholar