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# Delay differential equation

About: Delay differential equation is a research topic. Over the lifetime, 7557 publications have been published within this topic receiving 163233 citations.

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14 Oct 1993

TL;DR: The present book builds upon the earlier work of J. Hale, "Theory of Functional Differential Equations" published in 1977 and attempts to maintain the spirit of that book and have retained approximately one-third of the material intact.

Abstract: The present book builds upon the earlier work of J. Hale, "Theory of Functional Differential Equations" published in 1977. The authors have attempted to maintain the spirit of that book and have retained approximately one-third of the material intact. One major change was a completely new presentation of linear systems (Chapter 6-9) for retarded and neutral functional differential equations. The theory of dissipative systems (Chapter 4) and global attractors was thoroughly revamped as well as the invariant manifold theory (Chapter 10) near equilibrium points and periodic orbits. A more complete theory of neutral equations is presented (Chapters 1,2,3,9,10). Chapter 12 is also entirely new and contains a guide to active topics of research. In the sections on supplementary remarks, the authors have included many references to recent literature, but, of course, not nearly all, because the subject is so extensive.

6,844 citations

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01 May 1989

TL;DR: Impulsive differential equations, that is, differential equations involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems.

Abstract: Many evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly. These processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the process. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. It is known, for example, that many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics and frequency modulated systems, do exhibit impulsive effects. Thus impulsive differential equations, that is, differential equations involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems.

3,856 citations

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02 Feb 2012

TL;DR: Delay Differential Equations as mentioned in this paper are a generalization of delay differential equations and have been used in a variety of applications in population dynamics, such as global stability for single species models and multi-species models.

Abstract: Delay Differential Equations: Introduction. Basic Theory of Delay Differential Equations. Characteristic Equations. Applications in Population Dynamics: Global Stability for Single Species Models. Periodic Solutions, Chaos, Stage Structures, And State Dependent Delays in Single Species Models. Global Stability for Multi-Species Models. Periodic Solutions in Multi-Species Models. Permanence. Neutral Delay Models. References. Appendix. Index.

3,192 citations

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01 Mar 1995

TL;DR: Monotone dynamical systems Stability and convergence Competitive and cooperative differential equations Irreducible cooperative systems Cooperative systems of delay differential equations Nonquasimonotone delay differential equation Quasimonoteone systems of parabolic equations A competition model Appendix Bibliography as discussed by the authors

Abstract: Monotone dynamical systems Stability and convergence Competitive and cooperative differential equations Irreducible cooperative systems Cooperative systems of delay differential equations Nonquasimonotone delay differential equations Quasimonotone systems of parabolic equations A competition model Appendix Bibliography.

2,282 citations

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31 Mar 1992

TL;DR: The Delay Logistic Equation (DLE) as mentioned in this paper is a delay-induced Bifurcation to Periodicity (DBE) model for deterministic linear systems.

Abstract: 1. The Delay Logistic Equation. 2. Delay Induced Bifurcation to Periodicity. 3. Methods of Linear Analysis. 4. Global Attractivity. 5. Models of Neutral Differential Systems. References. Index.

2,007 citations