Nonautonomous finite-time dynamics. (English) Zbl 1148.37010

Berger, Arno; Son, Doan Thai; Siegmund, Stefan

Discrete Contin. Dyn. Syst., Ser. B 9, No. 3-4, 463-492 (2008).

Summary: Nonautonomous differential equations on finite-time intervals play an increasingly important role in applications that incorporate time-varying vector fields, e.g. observed or forecasted velocity fields in meteorology or oceanography which are known only for times \(t\) from a compact interval. While classical dynamical systems methods often study the behaviour of solutions as \(t\to\pm\infty\), the dynamic partition (originally called the EPH partition) aims at describing and classifying the finite-time behaviour. We discuss fundamental properties of the dynamic partition and show that it locally approximates the nonlinear behaviour. We also provide an algorithm for practical computations with dynamic partitions and apply it to a nonlinear three-dimensional example.

MSC:

37B55	Topological dynamics of nonautonomous systems
37D05	Dynamical systems with hyperbolic orbits and sets
37D10	Invariant manifold theory for dynamical systems
37N10	Dynamical systems in fluid mechanics, oceanography and meteorology
86A05	Hydrology, hydrography, oceanography
86A10	Meteorology and atmospheric physics

Keywords:

hyperbolicity; nonautonomous differential equations on finite-time intervals; dynamic partition