Symmetries and Semi-invariants in the Analysis of Nonlinear Systems
Symmetries and Semi-invariants in the Analysis of Nonlinear Systems details the analysis of continuous- and discrete-time dynamical systems described by differential and difference equations respectively. Differential geometry provides the essential tools
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Laura Menini Antonio Tornambè
Symmetries and Semi-invariants in the Analysis of Nonlinear Systems
Prof. Laura Menini Dipto. Informatica Sistemi e Produzione Università di Roma-Tor Vergata Via del Politecnico 1 00133 Rome Italy [email protected]
Prof. Antonio Tornambè Dipto. Informatica Sistemi e Produzione Università di Roma-Tor Vergata Via del Politecnico 1 00133 Rome Italy [email protected]
ISBN 978-0-85729-611-5 e-ISBN 978-0-85729-612-2 DOI 10.1007/978-0-85729-612-2 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2011928506 Mathematics Subject Classification: 93C15, 93C55, 34C14, 34C20, 93D30, 93B18, 93B27 © Springer-Verlag London Limited 2011 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of 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 laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: VTeX UAB, Lithuania Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
The goal of this book is to present several concepts useful for the analysis of dynamical systems, and to illustrate, in the last two chapters, how they can be actually applied to improve the state of the art for two classical topics in nonlinear systems theory: the linearization of a nonlinear system by state immersion and the study of stability of equilibrium points. The main reasoning that led us to writing this book is that some concepts that are already well developed in the literature become more important if presented together. Three of such concepts are homogeneity, symmetries (and orbital symmetries for continuous-time systems) and Lie algebras, which, in our opinion, can be better understood if symmetries are seen as a generalization of homogeneity, and Lie algebras (seen as generators of Lie groups) as a generalization of symmetries. Another very well known concept is that of first integral, that is particularly helpful for researchers working on Hamiltonian systems, or on stability of switched systems. In our opinion, similar attention should be paid to the generalization of first
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