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Nonautonomous Difference Equations

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2002)

Abstract

The long-time behavior of (autonomous) difference equations or discrete semidynamical systems exhibits various features, which occur in their continuous counterpart of (evolutionary) differential equations only for higher or even infinite-dimensional state spaces. This has various reasons, like absence of monotone solutions, lack of existence and uniqueness of backward solutions, or topological deficits caused by solution sequences rather than connected curves. As a consequence, besides further intricacies there are one-dimensional chaotic maps, there is no Poincaré–Bendixson theory (cf., e.g., [9, p. 333, Theorem (24.6)]) for planar maps, no limit set dichotomy for monotone maps (see [212]) or a comparatively subtle linearization theory for discrete systems, which are not homeomorphisms (see also Chap. 5).

Keywords

  • Difference Equation
  • Kutta Method
  • Global Attractor
  • Discrete Interval
  • Forward Solution

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Correspondence to Christian Pötzsche .

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Pötzsche, C. (2010). Nonautonomous Difference Equations. In: Geometric Theory of Discrete Nonautonomous Dynamical Systems. Lecture Notes in Mathematics(), vol 2002. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14258-1_2

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