SPRINGER BRIEFS IN MATHEMATICS Xiaoying Han Peter Kloeden Attractors Under Discretisation 123
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Xiaoying Han Peter Kloeden Attractors Under Discretisation 123
Xiaoying Han Department of Mathematics and Statistics Auburn University Auburn, AL USA Peter Kloeden School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan, Hubei China ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-319-61933-0 ISBN 978-3-319-61934-7 (ebook) DOI 10.1007/978-3-319-61934-7 Library of Congress Control Number: 2017947864 The Author(s) 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Dedicated to the memory of Karin Wahl-Kloeden 1954 2016
Preface Numerical dynamics is concerned with how well a numerical scheme applied to a differential equation replicates the dynamical behaviour of the dynamical system generated by the differential equation, in particular its long-term or asymptotic behaviour. This essentially involves the comparison of the dynamical behaviour of a continuous-time dynamical system with that of a corresponding discrete-time dynamical system. There are two broad classes of systems of particular interest: dissipative systems, which have an attractor, and non-dissipative systems, such as Hamiltonian systems, which preserve some structural feature or quantity. This work focusses on the preservation of attractors and saddle points of ordinary differential equations under discretisation. Key results for autonomous ODEs were obtained in the 1980s by Beyn for saddle points and Kloeden and Lorenz for attractors. One-step numerical schemes with a constant step size were considered, so the resulting discrete-time dynamical system was also autonomous. Autonomous dynamical systems with a saddle point may not be dissipative, but the results are nevertheless relevant for dissipative systems as they apply to what may happen inside an attractor. The theory of non-autonomous dynamical systems has undergone intensive development during the past 20 years, with the introduction of two kinds of non-autonomous attractor: pullback and forward attractors. In principle, a non-autonomous dynamical system can vary quite arbitrarily in time, but to obtain approximation results some sort of uniformity is required. One of the main aims of this book is to present new results on the discretisation of dissipative non-autonomous dynamical systems that have been obtained in recent years, in particular work on the properties of non-autonomous omega limit sets and their approximations by numerical schemes. These results are also of interest for autonomous dynamical systems that are approximated by a numerical scheme with variable time steps, and thus by a discrete-time non-autonomous dynamical system. The emphasis here is on the finite-dimensional case, i.e. on ordinary differential equations. Some similar results are known for the infinite dimensional case, e.g., vii
viii Preface systems generated by partial differential equations, but this case requires more sophisticated technical tools. The autonomous part of this book is based on lecture notes given over many years by the second author in Frankurt am Main and later in Wuhan. The non-autonomous part is much more recent and is based on papers published in various mathematical journals. Auburn, Wuhan May 2017 Xiaoying Han Peter Kloeden
Contents Part I Dynamical Systems and Numerical Schemes 1 Lyapunov Stability and Dynamical Systems... 3 1.1 Lyapunov Stability... 3 1.2 Autonomous Dynamical Systems... 6 1.3 Invariant Sets... 7 1.4 Limit Sets... 8 1.5 Attractors... 9 2 One Step Numerical Schemes... 11 2.1 Discretisation Error.... 12 2.2 General One Step Schemes.... 13 2.2.1 Taylor Schemes... 14 2.2.2 Schemes Derived by Integral Approximations... 16 2.3 Orders of Local and Global Convergence.... 21 2.4 Consistency... 23 2.5 Numerical Instability.... 25 2.6 Steady States of Numerical Schemes... 27 Part II Steady States Under Discretisation 3 Linear Systems... 35 3.1 Linear ODE in R 1... 36 3.2 Linear ODE in C 1... 37 3.3 The General Linear Case... 38 4 Lyapunov Functions... 41 4.1 Linear Systems Revisited... 43 4.2 Application: The Linear Euler Scheme... 44 4.3 Application: The Nonlinear Euler Scheme... 45 ix
x Contents 5 Dissipative Systems with Steady States... 49 6 Saddle Points Under Discretisation... 55 6.1 Saddle Points and the Euler Scheme.... 60 6.1.1 A Nonlinear Example... 61 6.1.2 Shadowing.... 63 6.2 General Case: Beyn s Theorem... 64 Part III Autonomous Attractors Under Discretisation 7 Dissipative Systems with Attractors... 69 7.1 Euler Scheme Dynamics... 70 7.2 Convergence of the Numerical Attractors... 73 8 Lyapunov Functions for Attractors... 77 8.1 Lyapunov Stability of Sets... 80 8.2 Yoshizawa s Theorem.... 81 9 Discretisation of an Attractor: General Case.... 83 Part IV Nonautonomous Limit Sets Under Discretisation 10 Dissipative Nonautonomous Systems... 91 10.1 Nonautonomous Omega Limit Sets... 92 10.2 Asymptotic Invariance... 93 10.2.1 Asymptotic Positive Invariance... 94 10.2.2 Asymptotic Negative Invariance... 95 11 Discretisation of Nonautonomous Limit Sets... 99 11.1 The Implicit Euler Scheme... 100 11.2 Upper Semi Continuous Converence of the Numerical Omega Limit Sets.... 101 12 Variable Step Size Discretisation of Autonomous Attractors... 105 12.1 Variable Time Step Limit Sets... 106 12.2 Upper Semi Continuous Convergence of the Numerical Omega Limit Sets.... 107 13 Discretisation of a Uniform Pullback Attractor.... 111 Notes... 119 References... 121
About the Authors Xiaoying Han obtained her Ph.D. from the University at Buffalo, USA in 2007 and is currently Professor at Auburn University, USA. Her main research interests are in random and non-autonomous dynamical systems and their applications. In addition to mathematical analysis of dynamical systems, she is also interested in the modelling and simulation of applied dynamical systems in biology, chemical engineering, ecology, material sciences, etc. Professor Han is a co-author of the books Applied Nonautonomous and Random Dynamical Systems (with T. Caraballo) published in the SpringerBrief series and Random Ordinary Differential Equations and Their Numerical Solutions (with P.E. Kloeden) published by Springer. Peter Kloeden completed his Ph.D. and D.Sc. at the University of Queensland, Australia in 1975 and 1995. He is currently Professor of Mathematics at Huazhong University of Science and Technology in China, and Affiliated Professor at Auburn University, USA. He has wide interests in the applications of mathematical analysis, numerical analysis, stochastic analysis, and dynamical systems. Professor Kloeden is a co-author of several influential books on non-autonomous dynamical systems, metric spaces of fuzzy sets, and in particular Numerical Solutions of Stochastic Differential equations (with E. Platen) published by Springer in 1992. He is a Fellow of the Society of Industrial and Applied Mathematics and was awarded the W.T. & Idalia Reid Prize in 2006. His current interests focus on non-autonomous and random dynamical systems and their applications in the biological sciences. xi
Part I Dynamical Systems and Numerical Schemes
Chapter 1 Lyapunov Stability and Dynamical Systems Abstract An introduction of Lyapunov stability and dynamical systems. First the concepts of stability, instability, attractivity, and asymptotic stability are introduced. Then autonomous semi-dynamical systems and their invariant sets, omega limit sets and attractors are defined. Keywords Steady state Lyapunov stability Semi-dynamical system Invariant set Omega limit set Attractor Consider an autonomous ordinary differential equation (ODE) with a steady state solution x(t) x for all t, i.e., dx dt = f (x), f (x ) = 0. (1.1) Recall that the solution of an autonomous ODE with an initial condition x(t 0 ) = x 0 satisfies the property x(t; t 0, x 0 ) x(t t 0 ; 0, x 0 ). Thus we can always take t 0 = 0 and write the corresponding solution simply as x(t; x 0 ). Essentially, this says that an autonomous system depends only on the elapsed time since starting and not on the actual values of the starting and current times. 1.1 Lyapunov Stability Definition 1.1 (Lyapunov Stability of a steady state solution) A steady state x of an autonomous ODE is said to be stable (in the sense of Lyapunov), if for every ε> 0 there exists a δ = δ(ε) > 0 such that for every initial value x 0 with x 0 x <δ the solution x(t; x 0 ) exists for all t 0 and satisfies the estimate x(t; x 0 ) x <ε for all t 0. Otherwise the steady state x is said to be unstable. An illustration of a stable steady state is provided in Fig. 1.1. The Author(s) 2017 X. Han and P. Kloeden, Attractors Under Discretisation, SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-61934-7_1 3