Dynamical Systems Stability, Controllability and Chaotic Behavior
At the end of the nineteenth century Lyapunov and Poincaré developed the so called qualitative theory of differential equations and introduced geometric-topological considerations which have led to the concept of dynamical systems. In its present abstract
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Werner Krabs • Stefan Pickl
Dynamical Systems Stability, Controllability and Chaotic Behavior
Prof. Dr. Werner Krabs Department of Mathematics Technical University of Darmstadt Schlossgartenstr. 7 64289 Darmstadt Germany
Prof. Dr. Stefan Pickl Universität der Bundeswehr München Department of Computer Science Werner Heisenberg Weg 39 85577 Neubiberg-München Germany [email protected]
ISBN 978-3-642-13721-1 e-ISBN 978-3-642-13722-8 DOI 10.1007/978-3-642-13722-8 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010932600 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg, Germany Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Foreword
Reflecting modelling dynamical systems by mathematical methods can be enriched by philosophical categories. The following introduction might catch the reader’s interest concerning some interdisciplinary dimensions and completes the holistic approach. “It has been said– and I was among those saying it– that any theory of explanation worth its salt should be able to make good predictions. If good predictions could not be made, the explanation could hardly count as serious. (. . .) I now want to move on to my main point. There is, I claim, no major conceptual difference between the problems of explaining the unpredictable in human affairs and in non-human affairs. There are, it is true, many remarkable successes of prediction in the physical sciences of which we are all aware, but these few successes of principled science making principled predictions are, in many ways, misleading. (. . .) The point I want to emphasize is that instability is as present in purely physical systems as it is in those we think of as characteristically human. Our ability to explain but not predict human behavior is in the same general category as our ability to explain but not predict many physical phenomena. The underlying reasons for the inability to predict are the same. (. . .) The concept of instability which accounts for many of these failures is one of the most neglected concepts in philosophy. We philosophers have as a matter of practice put too
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