Lecture Notes in Mathematics

By: Christian Pötzsche

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Paperback | 18 September 2010

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432 Pages

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23.5 x 16.51 x 2.54

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Nonautonomous dynamical systems provide a mathematical framework for temporally changing phenomena, where the law of evolution varies in time due to seasonal, modulation, controlling or even random effects. Our goal is to provide an approach to the corresponding geometric theory of nonautonomous discrete dynamical systems in infinite-dimensional spaces by virtue of 2-parameter semigroups (processes). These dynamical systems are generated by implicit difference equations, which explicitly depend on time. Compactness and dissipativity conditions are provided for such problems in order to have attractors using the natural concept of pullback convergence. Concerning a necessary linear theory, our hyperbolicity concept is based on exponential dichotomies and splittings. This concept is in turn used to construct nonautonomous invariant manifolds, so-called fiber bundles, and deduce linearization theorems. The results are illustrated using temporal and full discretizations of evolutionary differential equations.

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"The book contains detailed information concerning the two-parameter semigroups defined by a quite general class of difference equations. ... The Hartman-Grobman theory also receives considerable attention. ... this is a well-written book which will be very useful to the reader interested in the topics which it discusses." (Russell A. Johnson, Mathematical Reviews, Issue 2012 a)

"The monograph is a rich resource for a consistent theory of nonautonomous difference equations, in particular their stability theory and the connection between linear and nonlinear systems. ... The reader ... who is interested in a thorough course on the theory of difference equations will benefit from this book which combines summaries on the different topics with precise and new results." (Jorg Harterich, Zentralblatt MATH, Vol. 1247, 2012)

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