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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Notes
- 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.
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Liu, X., Stechlinski, P. (2017). Basic Theory. In: Infectious Disease Modeling. Nonlinear Systems and Complexity, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-319-53208-0_1
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DOI: https://doi.org/10.1007/978-3-319-53208-0_1
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