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Published in:
Cover of the book

2017 | OriginalPaper | Chapter

1. Basic Theory

Authors : Xinzhi Liu, Peter Stechlinski

Published in: Infectious Disease Modeling

Publisher: Springer International Publishing

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Abstract

Necessary mathematical definitions and concepts are presented in this chapter. Fundamental theory of ordinary differential equations is given first, followed by stability theory (including the notion of partial stability). The theory of delay differential equations, impulsive systems, and stochastic differential equations are also highlighted, with an emphasis placed upon stability notions and methods.

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next chapter Hybrid and Switched Systems
Footnotes
1
All eigenvalues have negative real part.
 
2
An orbit is called isolated if there exists a neighborhood containing said orbit for which there exists no other periodic orbit. (This is not possible in linear ODE systems.)
 
3
A solution ϕ ≡ ϕ(⋅ ; x 0) of (1.2) is said to be a periodic if there exists T > 0 such that φ(t + T; x 0) = φ(t; x 0) for all time t. The smallest T for which this equality holds is called the period.
 
4
Complete normed vector space.
 
Literature
15.
D. Bainov, P. Simeonov, Impulsive Differential Equations: Asymptotic Properties of the Solutions (World Scientific, Singapore, 1995)
47.
A.F. Filippov, Differential Equations with Discontinuous Righthand Sides (Kluwer, Dordrecht, 1988)
56.
J.K. Hale, Ordinary Differential Equations (Robert E. Krieger, Florida, 1980)
57.
J.K. Hale, Partial Stability and Control (Birkhauser, Boston, 1998)
58.
J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations (Springer, New York, 1993)
71.
H.K. Khalil, Nonlinear Systems (Prentice Hall, Upper Saddle River, 2002)
76.
V. Lakshmikantham, D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations (World Scientific, Singapore, 1989)
77.
V. Lakshmikantham, S. Leela, Differential and Integral Equations (Academic Press, New York, 1969)
78.
V. Lakshmikantham, M.R.M. Rao, Theory of Integro-Differential Equations (Gordon and Breach, Amsterdam, 1995)
106.
X. Mao, Stochastic Differential Equations and Their Applications (Horwood, Chichester, 2007)
109.
J.D. Meiss, Differential Dynamical Systems (Society for Industrial and Applied Mathematics, Philadelphia, 2007)
123.
L. Perko, Differential Equations and Dynamical Systems (Springer, New York, 2001)
Metadata
Title
Basic Theory
Authors
Xinzhi Liu
Peter Stechlinski
Copyright Year
2017
Book
Infectious Disease Modeling

Print ISBN: 978-3-319-53206-6

Electronic ISBN: 978-3-319-53208-0

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-53208-0_1

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