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ot95 mortell fm-c.qxp 5/26/2005 4:54 PM Page 1 Singular Perturbations and Hysteresis ot95 mortell fm-c.qxp 5/26/2005 4:54 PM Page 2 ot95 mortell fm-c.qxp 5/26/2005 4:55 PM Page 3 Singular Perturbations and Hysteresis Edited by Michael P. Mortell University College Cork Cork, Ireland Robert E. O’Malley University of Washington Seattle, Washington Alexei Pokrovskii University College Cork Cork, Ireland Vladimir Sobolev Samara State University Samara, Russia Society for Industrial and Applied Mathematics Philadelphia ot95 mortell fm-c.qxp 5/26/2005 4:55 PM Page 4 Copyright © 2005 by the Society for Industrial and Applied Mathematics. 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Control Number: 2005927224 ISBN 0-89871-597-0 Royalties from the sale of this book are placed in a fund to help students attend SIAM meetings and other SIAM-related activities. This fund is administered by SIAM, and qualified individuals are encouraged to write directly to SIAM for guidelines. is a registered trademark. “main” (cid:105) (cid:105) 2005/3/6 page i (cid:105) (cid:105) Contents Preface ix 1 A Naive View of Time Relaxation and Hysteresis 1 A. Pokrovskii, V. Sobolev 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Stop and Play nonlinearities . . . . . . . . . . . . . 1 1.1.2 Netushil’s representation . . . . . . . . . . . . . . . 4 1.1.3 General principle . . . . . . . . . . . . . . . . . . . . 7 1.1.4 Closed loop systems . . . . . . . . . . . . . . . . . . 9 1.2 Hysteresis Phenomena . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.1 Why bother with equations with hysteresis?. . . . . 12 1.2.2 Generalities . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.3 Non-ideal relay . . . . . . . . . . . . . . . . . . . . . 16 1.2.4 The parallel connection of non-ideal relays . . . . . 19 1.2.5 Preisach model . . . . . . . . . . . . . . . . . . . . . 20 1.2.6 Controllable Preisach nonlinearities . . . . . . . . . 24 1.2.7 Minor loops . . . . . . . . . . . . . . . . . . . . . . . 26 1.2.8 Identification Principle . . . . . . . . . . . . . . . . 31 1.2.9 Closed loop systems with hysteresis . . . . . . . . . 33 1.3 Singular Perturbation Phenomena . . . . . . . . . . . . . . . . . 37 1.3.1 Initial layer . . . . . . . . . . . . . . . . . . . . . . . 37 1.3.2 Jump point . . . . . . . . . . . . . . . . . . . . . . . 38 1.3.3 Canard trajectory . . . . . . . . . . . . . . . . . . . 39 1.3.4 Relaxation oscillations and hysteresis-like behaviour 41 1.4 Hysteresis vs Time Relaxation . . . . . . . . . . . . . . . . . . . 44 1.4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . 44 1.4.2 Hysteresistechniqueintheanalysisofsingularlyper- turbed equations . . . . . . . . . . . . . . . . . . . . 47 1.4.3 Hysteresis phenomena as a source of new singularly perturbed problems . . . . . . . . . . . . . . . . . . 47 1.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 53 1.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 53 Bibliography 55 i (cid:105) (cid:105) (cid:105) (cid:105) “main” (cid:105) (cid:105) 2005/3/6 page ii (cid:105) (cid:105) ii Contents 2 Frustration Minimization, Hysteresis and the El Farol Problem 61 R. Cross, M. Grinfeld, H. Lamba, A. Pittock 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.2 The Arthur Approach . . . . . . . . . . . . . . . . . . . . . . . . 62 2.3 An Alternative Approach to Decision Making . . . . . . . . . . 63 2.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.3.2 Construction . . . . . . . . . . . . . . . . . . . . . . 64 2.3.3 Decision making . . . . . . . . . . . . . . . . . . . . 64 2.4 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.5 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 68 Bibliography 71 3 Hysteresis in Singularly Perturbed Problems 73 P. Krejˇc´ı 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2 Physical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3 Statement of the Problem and Main Results . . . . . . . . . . . 77 3.4 The Play Operator . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.5 Uniformly Bounded Oscillation. . . . . . . . . . . . . . . . . . . 83 3.6 Singularly Perturbed Equations . . . . . . . . . . . . . . . . . . 86 3.7 Stability of the Flow . . . . . . . . . . . . . . . . . . . . . . . . 94 3.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 97 Bibliography 99 4 Combined Asymptotic Expansions 101 E. Benoˆıt, A. Fruchard, A. El Hamidi 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2 Computations with Combined Asymptotic Expansions . . . . . 103 4.3 Main Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.4 Improvement to Turning Points with Canards . . . . . . . . . . 106 4.5 Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Bibliography 109 5 Contrast Structures of Alternating Type 111 A. Vasilieva 5.1 Equation without Explicit Dependence on u . . . . . . . . . . . 111 x 5.2 Equation with Weak Dependence on u . . . . . . . . . . . . . . 115 x 5.3 The Case when Degenerate Equation Has Several Roots . . . . 116 5.4 Periodic Contrast Structures of Step Type . . . . . . . . . . . . 117 5.5 Periodic Contrast Structures of Alternating Type . . . . . . . . 122 5.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 123 Bibliography 125 (cid:105) (cid:105) (cid:105) (cid:105) “main” (cid:105) (cid:105) 2005/3/6 page iii (cid:105) (cid:105) Contents iii 6 Multi-Dimensional Internal Layers 127 N. Nefedov 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.2 Step-type Contrast Structures in Partial Differential Equations. 128 6.2.1 The noncritical case . . . . . . . . . . . . . . . . . . 128 6.2.2 The case of balanced nonlinearity . . . . . . . . . . 133 6.3 Spike-type Contrast Structures in Partial Differential Equations 135 6.3.1 Moving spikes . . . . . . . . . . . . . . . . . . . . . 135 6.3.2 Stationary spikes . . . . . . . . . . . . . . . . . . . . 141 6.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.4.1 Phase transitions models . . . . . . . . . . . . . . . 143 6.4.2 Stationary spikes in Fisher’s equation . . . . . . . . 144 6.4.3 Moving spikes in Fisher’s equation . . . . . . . . . . 145 6.5 Asymptotic Method of Differential Inequalities . . . . . . . . . . 146 6.5.1 Basic ideas . . . . . . . . . . . . . . . . . . . . . . . 146 6.5.2 SLEP-method . . . . . . . . . . . . . . . . . . . . . 148 6.5.3 Outline of proof . . . . . . . . . . . . . . . . . . . . 149 6.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 150 Bibliography 151 7 Geometry of Singular Perturbations: Critical Cases 153 V. Sobolev 7.1 Introduction. Elements of the Geometric Theory of Singularly Perturbed Systems . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.1.1 Slow integral manifolds . . . . . . . . . . . . . . . . 153 7.1.2 Asymptotic representation of integral manifolds . . 156 7.1.3 Stability of slow integral manifolds . . . . . . . . . . 157 7.1.4 Critical cases . . . . . . . . . . . . . . . . . . . . . . 157 7.2 Singular Singularly Perturbed Systems . . . . . . . . . . . . . . 158 7.2.1 Existence of slow integral manifolds . . . . . . . . . 159 7.2.2 Explicit and implicit slow integral manifolds . . . . 161 7.2.3 Parametric representation of integral manifolds . . . 162 7.2.4 High-gain control. . . . . . . . . . . . . . . . . . . . 166 7.2.5 Asymptotics with fractional exponents of the small parameter . . . . . . . . . . . . . . . . . . . . . . . . 167 7.3 Systems with Slow Dissipation . . . . . . . . . . . . . . . . . . . 168 7.3.1 Gyroscopic systems . . . . . . . . . . . . . . . . . . 168 7.3.2 Precessional and nutational motions . . . . . . . . . 169 7.3.3 Vertical gyro with radial corrections . . . . . . . . . 173 7.3.4 Heavy gyroscope . . . . . . . . . . . . . . . . . . . 174 7.3.5 A one rigid-link flexible-joint manipulator . . . . . . 175 7.4 Branching of Slow Integral Manifolds . . . . . . . . . . . . . . . 178 7.4.1 Formulation of the problem. Preliminaries. . . . . . 178 7.4.2 Quasi-homogeneous degenerate equations . . . . . . 180 7.4.3 Homogeneous systems . . . . . . . . . . . . . . . . . 182 (cid:105) (cid:105) (cid:105) (cid:105) “main” (cid:105) (cid:105) 2005/3/6 page iv (cid:105) (cid:105) iv Contents 7.4.4 Quasi-polynomial degenerate equations . . . . . . . 183 7.4.5 Cheap control . . . . . . . . . . . . . . . . . . . . . 186 7.5 Appendix: Optimal Estimation in Gyroscopic Systems . . . . . 190 7.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 190 7.5.2 TheequationsofaKalmanfilterforgyroscopicsystems190 7.5.3 Precessional equations in the deterministic case . . . 193 7.5.4 Optimalfilteringintheprecessionalequationsofgy- roscopic systems . . . . . . . . . . . . . . . . . . . . 194 7.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 202 Bibliography 203 8 Black Swans and Canards in Laser and Combustion Models 207 E. Shchepakina, V. Sobolev 8.1 Integral Manifolds and Canards . . . . . . . . . . . . . . . . . . 214 8.1.1 Canards of two-dimensional systems . . . . . . . . . 215 8.1.2 Canards of three-dimensional systems . . . . . . . . 216 8.1.3 Asymptotic expansions for canards . . . . . . . . . . 219 8.2 The Stable/Unstable Slow Integral Manifolds. . . . . . . . . . . 224 8.2.1 Black swans. . . . . . . . . . . . . . . . . . . . . . . 224 8.2.2 Black swans and canards . . . . . . . . . . . . . . . 228 8.3 Chemical Models . . . . . . . . . . . . . . . . . . . . . . . . . . 229 8.3.1 Lang-Kobayashi equations. . . . . . . . . . . . . . . 229 8.3.2 Exchange of stability in a high–gain control problem 230 8.3.3 The simple laser . . . . . . . . . . . . . . . . . . . . 231 8.3.4 The classical combustion models . . . . . . . . . . . 231 8.3.5 Canard travelling waves . . . . . . . . . . . . . . . . 237 8.3.6 Gas combustion in a dust–laden medium . . . . . . 243 8.4 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 249 Bibliography 251 9 Multi-Scale Analysis of Pressure Driven Flames 257 V. Bykov, I. Goldfarb, V. Gol’dshtein 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 9.2 Problem Statement - General Description . . . . . . . . . . . . . 260 9.3 Method of Integral Manifolds (MIM) – Asymptotic Method of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 9.4 Linear Friction, No Inertia - Zel’dovich’s Approach . . . . . . . 264 9.4.1 Preheat sub-zone . . . . . . . . . . . . . . . . . . . . 265 9.4.2 Reaction sub-zone . . . . . . . . . . . . . . . . . . . 266 9.4.3 Flame velocity . . . . . . . . . . . . . . . . . . . . . 267 9.4.4 Comparison with numerics . . . . . . . . . . . . . . 268 9.4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 269 9.5 Linear Friction, No Inertia - MIM Approach . . . . . . . . . . . 270 9.5.1 Singularly perturbed system - SPS . . . . . . . . . . 271 (cid:105) (cid:105) (cid:105) (cid:105) “main” (cid:105) (cid:105) 2005/3/6 page v (cid:105) (cid:105) Contents v 9.5.2 Application of MIM . . . . . . . . . . . . . . . . . . 272 9.5.3 Flame velocity . . . . . . . . . . . . . . . . . . . . . 276 9.6 Non-linear Friction, No Inertia . . . . . . . . . . . . . . . . . . . 276 9.6.1 Singularly perturbed system - SPS . . . . . . . . . . 277 9.6.2 Application of MIM . . . . . . . . . . . . . . . . . . 278 9.6.3 Flame velocity . . . . . . . . . . . . . . . . . . . . . 278 9.6.4 Theory vs numerics . . . . . . . . . . . . . . . . . . 279 9.7 Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 9.7.1 Singularly perturbed system - SPS . . . . . . . . . 282 9.7.2 Flame front structure - sub-zones. . . . . . . . . . . 284 9.7.3 Matching point . . . . . . . . . . . . . . . . . . . . . 287 9.7.4 Flame velocity . . . . . . . . . . . . . . . . . . . . . 288 9.7.5 Theory vs numerics . . . . . . . . . . . . . . . . . . 289 9.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 9.9 Appendix - Recent Results . . . . . . . . . . . . . . . . . . . . 294 9.10 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Bibliography 297 10 Split-Hyperbolicity, Hysteresis and Lang-Kobayashi Equations 299 A. Pokrovskii, O. Rasskazov, R. Studdert 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 10.2 Split-Hyperbolicity in Product-Spaces . . . . . . . . . . . . . . . 300 10.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 10.4 Split-Hyperbolicity in the Analysis of Chaotic Behavior . . . . . 304 10.4.1 Maps strongly compatible with topological Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . . 304 10.4.2 Split-hyperbolicityintheanalysisofstrongcompat- ibility . . . . . . . . . . . . . . . . . . . . . . . . . . 309 10.5 Application to the Truncated Lang-Kobayashi Equations . . . . 311 10.5.1 Lang-Kobayashi equations. . . . . . . . . . . . . . . 311 10.5.2 Poincar´e map . . . . . . . . . . . . . . . . . . . . . . 312 10.5.3 Main theorem . . . . . . . . . . . . . . . . . . . . . 314 10.5.4 Idea of the proof of Theorem 10.5.2 . . . . . . . . . 315 10.6 Proof of Lemma 10.5.2 . . . . . . . . . . . . . . . . . . . . . . . 318 10.6.1 Auxiliary translation operator . . . . . . . . . . . . 318 10.6.2 Lipschitz continuity of the translation operator . . . 320 10.6.3 Numerical integration . . . . . . . . . . . . . . . . . 321 10.6.4 Proof of s-hyperbolicity . . . . . . . . . . . . . . . . 325 10.6.5 Computations . . . . . . . . . . . . . . . . . . . . . 335 10.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 337 Bibliography 339 Index 343 (cid:105) (cid:105) (cid:105) (cid:105) “main” (cid:105) (cid:105) 2005/3/6 page vi (cid:105) (cid:105) vi Contents (cid:105) (cid:105) (cid:105) (cid:105)

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Author:Michael P. Mortell, Robert E. O'Malley, Alexei Pokrovskii, Vladimir Sobolev
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