American Institute of Mathematical Sciences

Advanced
May  2008, 9(3&4, May): 463-492. doi: 10.3934/dcdsb.2008.9.463

Nonautonomous finite-time dynamics

Arno Berger 1, , Doan Thai Son 2, and  Stefan Siegmund 2,

1. 

Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
2. 

Department of Mathematics, Dresden University of Technology, 01062 Dresden, Germany, Germany

Received  February 2007 Revised  June 2007 Published  February 2008

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 3-dimensional example.
Keywords: nonautonomous differential equations on finite-time intervals, dynamic partition., Hyperbolicity.
Mathematics Subject Classification: Primary: 37D05, 37D10, 37B55; Secondary: 37N1.
Citation: Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463
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