Monotone dynamical systems. (English) Zbl 1094.34003

Hirsch, M. W.; Smith, Hal

Cañada, A.(ed.) et al., Ordinary differential equations. Vol. II. Amsterdam: Elsevier/North Holland (ISBN 0-444-52027-9/hbk). Handbook of Differential Equations, 239-357 (2005).

The paper, a part of the handbook of differential equations, is a substantial survey on dynamical systems in ordered spaces. The main focus is on the asymptotical behaviour of monotone dynamical systems. The paper contains seven chapters organized as follows:
0. Introduction; 1. Strongly order-preserving semiflows; 2. Generic convergence and stability; 3. Ordinary differential equations; 4. Delay differential equations; 5. Monotone maps; 6. Semilinear parabolic equations.
At the end of the paper, 248 references are listed, covering a big part of the existing literature on the subject.
For the entire collection see [Zbl 1074.34003].

Reviewer: Gerd Herzog (Karlsruhe)

MSC:

34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34C12	Monotone systems involving ordinary differential equations
34D05	Asymptotic properties of solutions to ordinary differential equations
34D09	Dichotomy, trichotomy of solutions to ordinary differential equations
34K12	Growth, boundedness, comparison of solutions to functional-differential equations
35K55	Nonlinear parabolic equations
47H07	Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces

Keywords:

order-preserving semiflows; generic convergence and stability; monotone maps; semilinear parabolic equations; quasimonotonicity