Principles of Discontinuous Dynamical Systems
Discontinuous dynamical systems have played an important role in both theory and applications during the last several decades. This is still an area of active research and techniques to make the applications more effective are an ongoing topic of interest
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Principles of Discontinuous Dynamical Systems Marat Akhmet Middle East Technical University, Ankara, Turkey
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Marat Akhmet Middle East Technical University Department of Mathematics 06531 Ankara, Turkey [email protected]
ISBN 978-1-4419-6580-6 e-ISBN 978-1-4419-6581-3 DOI 10.1007/978-1-4419-6581-3 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010930467 Mathematics Subject Classification (2010): 34-01, 34-02, 34A12, 34A37, 34C23, 34C25, 34C28, 34D09, 34D20, 34D45, 34E05, 37G15, 37D45, 37D99, 37N05 c Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To my beloved mother, wife and daughters: Zhanar and Laila
Preface
The main subject of this book is discontinuous dynamical systems. These have played an extremely important role theoretically, as well as in applications, for the last several decades. Still, the theory of these systems seems very far from being complete, and there is still much to do to make the application of the theory more effective. This is especially true of equations with trajectories discontinuous at moments that are not prescribed. The book is written not only on the basis of research experience but also, importantly, on the basis of the experience of teaching the course of Impulsive differential equations for about 10 years to the graduate students of mathematics. It is useful for a beginner as we try not to avoid any difficult instants in delivering the material. Delicate questions that are usually ignored in a research monograph are thoroughly addressed. The standard material on equations with fixed moments of impulses is presented in a compact and definitive form. It contains a large number of exercises, examples, and figures, which will aid the reader in understanding the enigmatic world of discontinuous dynamics. The following peculiarity is very important: the material is built on the basis of close parallelism with ordinary differential equations theory. For example, even higher order differentiability of solutions, which has never been considered before, is presented with a full definition and detailed proofs. At the same time, the definition of the derivatives as coefficients of the expansion is fruitfully used, which is very rare in
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