Nonautonomous finite-time dynamics

Abstract

Nonautonomous differential equations on finite-time intervals playan increasingly important role in applications that incorporatetime-varying vector fields, e.g. observed or forecasted velocityfields in meteorology or oceanography which are known only fortimes $t$ from a compact interval. While classical dynamicalsystems methods often study the behaviour of solutions as $t \to\pm\infty$, the dynamic partition (originally called the EPHpartition) aims at describing and classifying the finite-timebehaviour. We discuss fundamental properties of the dynamicpartition and show that it locally approximates the nonlinearbehaviour. We also provide an algorithm for practical computationswith dynamic partitions and apply it to a nonlinear 3-dimensionalexample.