Selected Titles in This Series

56 Jane Cronin and Robert E. O'Malley, Jr. , Editors, Analyzing multiscale phenomena using singular perturbation methods (Baltimore, Maryland, January 1998)

55 Frederick Hoffman, Editor, Mathematical aspects of artificial intelligence (Orlando,

Florida, January 1996)

54 Renato Spigler and Stephanos Venakides, Editors, Recent advances in partial

differential equations (Venice, Italy, June 1996)

53 David A. Cox and Bernd Sturmfels , Editors, Applications of computational algebraic

geometry (San Diego, California, January 1997)

52 V . Mandrekar and P. R. Masani , Editors, Proceedings of the Norbert Wiener

Centenary Congress, 1994 (East Lansing, Michigan, 1994)

51 Louis H. Kauffman, Editor, The interface of knots and physics (San Francisco,

California, January 1995)

50 Robert Calderbank, Editor, Different aspects of coding theory (San Francisco,

California, January 1995)

49 Robert L. Devaney , Editor, Complex dynamical systems: The mathematics behind the

Mandlebrot and Julia sets (Cincinnati, Ohio, January 1994)

48 Walter Gautschi, Editor, Mathematics of Computation 1943-1993: A half century of

computational mathematics (Vancouver, British Columbia, August 1993)

47 Ingrid Daubechies , Editor, Different perspectives on wavelets (San Antonio, Texas,

January 1993)

46 Stefan A. Burr, Editor, The unreasonable effectiveness of number theory (Orono,

Maine, August 1991)

45 D e W i t t L. Sumners , Editor, New scientific applications of geometry and topology

(Baltimore, Maryland, January 1992)

44 Be la Bollobas, Editor, Probabilistic combinatorics and its applications (San Francisco,

California, January 1991)

43 Richard K. Guy, Editor, Combinatorial games (Columbus, Ohio, August 1990)

42 C. Pomerance , Editor, Cryptology and computational number theory (Boulder,

Colorado, August 1989)

41 R. W . Brockett , Editor, Robotics (Louisville, Kentucky, January 1990)

40 Charles R. Johnson, Editor, Matrix theory and applications (Phoenix, Arizona,

January 1989)

39 Robert L. Devaney and Linda Keen, Editors, Chaos and fractals: The mathematics

behind the computer graphics (Providence, Rhode Island, August 1988)

38 Juris Hartmanis , Editor, Computational complexity theory (Atlanta, Georgia, January

1988)

37 Henry J. Landau, Editor, Moments in mathematics (San Antonio, Texas, January 1987)

36 Carl de Boor, Editor, Approximation theory (New Orleans, Louisiana, January 1986)

35 Harry H. Panjer, Editor, Actuarial mathematics (Laramie, Wyoming, August 1985)

34 Michael Anshel and Wil l iam Gewirtz , Editors, Mathematics of information

processing (Louisville, Kentucky, January 1984)

33 H. P e y t o n Young, Editor, Fair allocation (Anaheim, California, January 1985)

32 R. W . McKelvey , Editor, Environmental and natural resource mathematics (Eugene,

Oregon, August 1984)

31 B . Gopinath, Editor, Computer communications (Denver, Colorado, January 1983)

30 S imon A. Levin, Editor, Population biology (Albany, New York, August 1983)

29 R. A. DeMil lo , G. I. Davida, D . P. Dobkin, M. A. Harrison, and R. J. Lipton, Applied cryptology, cryptographic protocols, and computer security models (San Francisco,

California, January 1981)

28 R. Gnanadesikan, Editor, Statistical data analysis (Toronto, Ontario, August 1982)

(Continued in the back of this publication)

http://dx.doi.org/10.1090/psapm/056

AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures

published as a subseries of Proceedings of Symposia in Applied Mathematics

The authors wish to dedicate this volume to the memory of William A. Harris, Jr., 1930-1998.

Bill's life and example motivate us to attempt to asymptotically approach his generosity of spirit, his enthusiasm for teaching and learning, and his persistent drive to analyze and understand differ­ential equations and their solutions.

Proceedings of Symposia in

APPLIED MATHEMATICS

Volume 56

Analyzing Multiscale Phenomena Using Singular Perturbation Methods

American Mathematical Society Short Course January 5-6, 1998 Baltimore, Maryland

Jane Cronin Robert E. O'Malley, Jr. Editors

& American Mathematical Society | Providence, Rhode Island

°^VDED

Editor ial Board

Marsha J. Berger Peter S. Constantin (Chair) Eitan Tadmor

LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE

SINGULAR PERTURBATION CONCEPTS OF DIFFERENTIAL EQUATIONS HELD IN BALTIMORE, MARYLAND

JANUARY 5-6, 1998

T h e AMS Short Course Series is sponsored by the Society's P rog ram Commi t t ee for National Meetings. T h e series is under the direction of t he Short Course

Subcommi t t ee of the P rogram Commi t t ee for Nat ional Meetings.

1991 Mathematics Subject Classification. P r imary 34D15, 34E15, 34E20, 35B25.

Library of Congress Cataloging- in-Publ icat ion Data

Analyzing multiscale phenomena using singular perturbation methods: American Mathematical Society short course, January 5-6, 1998, Baltimore, Maryland / Jane Cronin, Robert E. O'Malley, Jr., editors.

p. cm.-(Proceedings of symposia in applied mathematics, ISSN 0160-7634; v. 56. AMS short course series)

Includes bibliographical references and index. ISBN 0-8218-0929-6 (alk. paper) 1. Singular perturbations (Mathematics) Congresses. 2. Differentiable dynamical systems.

I. Cronin, Jane, 1922- . II. O'Malley, Robert E. III. Series: Proceedings of symposia in applied mathematics; v. 56. IV. Series: Proceedings of symposia in applied mathematics. AMS short course lecture notes. QA372.A54 1999 515 ,.35-dc21 99-13036

CIP

Copying and reprinting. Material in this book may be reproduced by any means for edu­cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg­ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission@ams.org.

Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.)

© 1999 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at URL: http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 04 03 02 01 00 99

Contents

Preface ix

Figuring out singular perturbations after a first course in ODEs ROBERT E. O 'MALLEY, J R . 1

The method of multiple scales MARK H. HOLMES 23

Computational methods for singularly perturbed systems SLIMANE ADJERID, MOHAMMED AIFFA, AND JOSEPH E. FLAHERTY 47

An introduction to geometric methods and dynamical systems theory for singular perturbation problems TASSO J. KAPER 85

Analysis of cellular oscillations JANE CRONIN 133

Exponential asymptotics and convection-diffusion-reaction models MICHAEL J. WARD 151

Index 185

Preface

An increasingly useful understanding of various singular perturbation phenom­ena has been achieved over the last fifty years. The basis of such progress grew from the revolutionary work of Poincare and Prandtl a century ago and, to a lesser extent, later efforts of van der Pol and of Kryloff and Bogoliuboff. Poincare, moti­vated by problems in celestial mechanics, introduced the definition of an asymptotic approximation (as a truncation of a divergent series) and realized that different ap­proximate solutions were needed on expanding time domains. Prandtl likewise introduced the idea of a boundary layer of nonuniform convergence to explain the paradoxes of viscous fluid flow. This explained aerodynamic lift, but has also had equally important applications throughout science, van der Pol studied relaxation oscillators occurring in electrical circuits, physiology, and even music by introducing a harmonic representation with slowly-varying coefficients, while Kryloff and Bo­goliuboff developed averaging methods to describe long-time oscillatory motion. A brief history is the subject of an appendix to O'Malley (1991) and a more thorough scholarly examination is certainly needed.

Singular perturbations can, somewhat naively, be characterized by the asymp­totic methods, such as matched asymptotic expansions and multiscale techniques, used to tame them. Increasingly, however, new applications require hybrid methods, partly analytical and partly numerical. Recently, concepts from dynamical systems theory, using invariant manifolds and various geometrical insights, are becoming critical to gaining deeper understanding. Just when we were too comfortable us­ing a few terms of an asymptotic approximation to describe solutions, new levels of precision (i.e., exponential asymptotics or asymptotics beyond all orders) have been shown to be necessary. Thus the familiar succession continues: challenging applications, the breakdown of traditional methods, progress utilizing new and old ideas, and unanticipated opportunities to apply new tools just developed.

The authors of the following papers seek to introduce you to singular pertur­bations as a mathematical subject arising from applications. Their insights reflect their individual experiences and expertise. Some important related topics aren't covered; for example, biofluiddynamics or chemically-reacting flows, stochastics and turbulence, numerical algorithms for stiff differential equations or those with rapidly-oscillating solutions, etc. Nonetheless, we hope that readers will be able to employ the methods described and the implicit philosophy suggested to learn about tomorrow's problems involving vastly different time and length scales.

The editors thank their good-natured colleagues and the AMS staff for their splendid cooperation in producing the short course and this proceedings volume.

ix

X PREFACE

References

N. N. Bogoliubov and Y. A. Mitropolsky (1961), Asymptotic Methods in the Theory of Non-linear Oscillations, Hindustan Publishing Co., Delhi.

J. Grasman (1987), Asymptotic Methods for Relaxation Oscillations and Ap­plications, Springer-Verlag, New York.

N. Kryloff and N. Bogoliuboff (1943), Introduction to Non-linear Mechanics, Princeton University Press, Princeton.

R. E. O'Malley, Jr. (1991), Singular Perturbation Methods for Ordinary Dif­ferential Equations, Springer-Verlag, New York.

Jane Cronin and Bob O'Malley December 1998

Index

A posteriori error estimates, 80 A posteriori estimates of discretization

errors, 47, 73 Aboufadel, 134, 143 accuracy of a perturbation expansion, 31 activator-inhibitor system, 152 adaptive approaches, 47 adaptive enrichment, 80 adaptive finite element software, 72 Adrian and Peachey, 133 Allen-Cahn equation, 172, 173, 174, 177 asymptotic, 151 asymptotic expansion, 4 asymptotic remainders, 6 asymptotic solution, 11 asymptotic solutions of difference equations,

37 asymptotic stability, 9 asymptotically negligible, 2 asymptotically stable slow manifolds, 87 attractive, 16

backward difference software DASSL, 73 balancing, 28, 33, 35 basepoints, 94 basin of attraction, 9 Beeler and Reuter, 133 Belousov-Zhabotinski reaction, 134 Bernoulli equation, 7 bifurcation equation, 149 bistability, 35 boundary function method, 6 boundary layer resonance, 151 boundary layer theory, 2 boundary layers, 7 bounding solutions, 7 bump function, 109 Burgers equation, 21, 63, 152, 158, 161, 163,

164, 166, 171

Cahn-Hilliard, 173, 174 Cahn-Hilliard equation, 177 canard, 15, 18 Carpenter, 134 Carrier-Pearson (CP) problem, 170 cell Peclet number, 50, 57 center manifold, 96

characteristic curve, 19 Cole-Hopf transformation, 158, 162 compact manifolds, 88 composite asymptotic series, 4 composite expansion, 5 constrained Allen-Cahn equation, 173, 174,

177 convection-diffusion equation, 68 convection-diffusion problems, 47 corner layer, 17 coupled mechanical oscillators, 85

DAE system, 176, 177 damped oscillator, 25 DASSL, 63 delayed bifurcation, 18 dichotomy, 7 dichotomy principal, 73 differential-algebraic problem, 10 differential-algebraic systems, 73 DiPrancesco and Noble, 133 disorder, 16 distinguished limits, 38 domain of attraction, 9 dormant solution, 15 Duffing oscillator, 120

eigenvalue problem, 153, 155, 165, 166, 168, 171, 173, 176, 178

eliminating secular terms, 5 entrainment of frequency, 137, 146 enzyme kinetics, 85 error estimates, 73 error function, 16 exchange of stability, 18 exponential asymptotics, 20 exponential contraction, 98 exponential ill-conditioning, 151, 155, 157,

164, 165, 168 exponential sensitivity, 157 exponentially small, 154, 156, 157, 160, 164,

166, 169, 175, 176 exponentially small eigenvalue, 77 exponentially small eigenvalues, 151-154, 157,

165, 178-180 exponentially small terms, 151, 153-155, 164,

166, 168, 171, 181

185

186

extreme sensitivity, 164

fast fibers, 94 fast stable fiber, 102 fast time variable, 4 fast unstable fiber, 101 Fenichel coordinates, 110 Fenichel normal form, 109, 111, 114, 115 Fenichel theory, 85, 107, 134, 141 Field-Noyes model, 134 finite difference, 47 finite element techniques, 47 flame propagation, 158 flame-front, 166 flame-front interfaces, 166-168, 170 Flanders and Stoker, 134, 143 forced Burgers equation, 162 Frankenhauser and Huxley, 133 fundamental solution, 3

Galerkin least-squares methods, 77 Galerkin least-squares stabilization, 48 Gauss-Legendre quadrature, 55 generalized asymptotic expansion, 4 geometric singular perturbation theory, 85 geometric theory, 13 Gierer Meinhardt model, 178 gradient Galerkin least-squares methods, 77 Green's function, 55

h-refinement, 47, 72 heteroclinic orbit, 17 hierarchical basis, 49 high-order methods, 48 Hodgkin and Huxley, 133 Hodgkin-Huxley equations, 134, 138 Hodgkin-Huxley theory, 133 homogenization, 5, 45 hp-refinement, 48

H'in's finite difference scheme, 52 implicit function theorem, 10 indeterminacy, 151, 155, 168, 171, 179 initial layer correction, 5, 11 initial layer subsystem, 10 inner expansion, 6, 97 inner variable, 4 integration by parts, 3 invariant manifolds, 13, 85, 98 invariant set, 88

Jacobian, 8 Jones, C.K.R.T., 85 jumping duck, 18

Kaper, 134

LaSalle, 134, 143 Legendre points, 59 Legendre polynomial, 50, 57

INDEX

Levinson, 11, 134 limit cycle, 19 linear turning point problems, 151 Lobatto quadrature, 48, 55, 67, 68 locally-invariant sets, 88

manifold with boundary, 89 MAPLE, 4, 56 matched asymptotic expansion, 112, 151, 153,

164, 168, 171 matching, 7 maximum principle, 20 McAllister, Noble, and Tsien, 133 metastability, 155, 158, 161, 163, 166, 170,

182 metastable, 164, 167, 168, 174, 176, 178,

180-182 metastable behavior, 151, 157, 158, 167 metastable dynamics, 158, 160, 168, 170 metastable phenomena, 155 metastable problem, 157 metastable viscous shock problems, 164 method of averaging, 30 method of matched asymptotic expansions,

96, 112, 164 method of multiple scales, 23 method of strained coordinates, 32 method-of-lines, 73 Mishchenko et al., 134, 145 Mishchenko and Rozov, 134, 145 morphogenesis, 152, 177, 178 Morse oscillator, 32 multi-bump homoclinic orbits, 122 multiple scale, 5 multiple-scale expansion, 27 multitime expansion, 4

Nipp, 134 Noble, 133, 134 Noble and Noble, 133 nonlocal eigenvalue problem, 179 nonlocal reaction diffusion equation, 152 nonlocal reaction-diffusion, 178 nonuniform convergence, 9 normal hyperbolicity, 85 normally hyperbolic, 134, 141 normally hyperbolic invariant manifold, 90,

120

O'Malley, 134 outer expansion, 11, 97 outer limit, 2 outer solution, 4 overlap, 7 overlap domain, 118

p-refinement, 47, 72 pacemaker impulses, 134, 135 parabolic and elliptic problems, 48

INDEX 187

passage through resonance, 37, 44 Peclet number, 49 Petrov-Galerkin, 51 Petrov-Galerkin methods, 48 phase asymptotically stable, 144 phase separation, 170 phase separation models, 152, 173 Pontryagin and Mishchenko, 134 power series, 3 projection, 156 projection method, 151, 152, 158, 161, 164,

167, 171, 175, 178 Purkinje fiber, 134

quadrature, 48, 55 quadtree-structured mesh, 73 quasi-steady state manifold, 11 quasiexponentials, 5

r-refinement, 47, 72 Radau integration, 48 Radau quadrature, 48, 55, 68 reaction-diffusion problems, 48 rectifier, 18 reduced equation, 2 regular expansion, 25 regular perturbation, 8 related phase separation models, 173 relaxation oscillations, 134 repulsive, 16 rescaling, 16 resonance, 32, 137 resonance bands, 119 resonance problem, 75 resonant sloshing, 91 Reynolds number, 49 Ricatti equation, 16

secular term, 26, 28, 38, 43 shape function, 55 shift-scaling transformations, 17 shock layer, 19, 20, 158, 160, 161, 163, 164,

166, 170 shock-type solutions, 152 shooting parameter, 14 Sibuya, 134 Silnikov orbits, 123 singular perturbation, 9 singularly perturbed boundary value prob­

lems, 151 slow manifold, 11, 95 slow manifolds, 92, 100, 114 slow subsystem, 10 slow time, 4 slow-fast system, 11 slowly-varying pendulum, 92, 106 Sobolev space, 49 solvability condition, 166, 170

solvability conditions, 152, 154, 172, 173, 176, 182

spike solutions, 178, 182 spike-type internal layers, 151 stability problem, 9 stable manifold, 7 stable manifold theorem, 89 stable matrix, 2 stiff computations, 2 stiff ordinary differential systems, 73 Stoker, 134, 143 streamwise upwinding, 77 stretching, 2 strong nonlinearities, 42 Sturm-Liouville eigenvalue problem, 159, 160,

162 super-stability, 17 superconvergence, 73 supersensitivity, 20 symmetric double-well potential, 163 synchronization, 135, 137

Taylor polynomial, 3 tensor product, 70 terminal layer, 7 Tikhonov, 11 Tikhonov-Levinson theory, 87 transition layer, 18 Traub and Miles, 133 turning point problems, 155, 156 turning points, 142 two-point problem, 14 two-timing, 5, 27

uniformly valid, 31 upper and lower solutions, 20 upwind-differencing, 2

van der Pol, 19, 163 van der Pol equation, 134, 138 variation-of-constants, 3 viscous Cahn-Hilliard equation, 173, 174 viscous shock, 164 viscous shock problems, 158, 163, 170

weakly coupled oscillators, 36 Winslow et al., 133 WKB method, 32

Yanagihara et al., 133

Selected Titles in This Series (Continued from the front of this publication)

27 L. A. Shepp, Editor, Computed tomography (Cincinnati, Ohio, January 1982)

26 S. A. Burr, Editor, The mathematics of networks (Pittsburgh, Pennsylvania, August 1981)

25 S. I. Gass, Editor, Operations research: mathematics and models (Duluth, Minnesota, August 1979)

24 W . F. Lucas, Editor, Game theory and its applications (Biloxi, Mississippi, January 1979)

23 R. V. Hogg, Editor, Modern statistics: Methods and applications (San Antonio, Texas, January 1980)

22 G. H. Golub and J. Oliger, Editors, Numerical analysis (Atlanta, Georgia, January 1978)

21 P. D . Lax, Editor, Mathematical aspects of production and distribution of energy (San Antonio, Texas, January 1976)

20 J. P. LaSalle, Editor, The influence of computing on mathematical research and

education (University of Montana, August 1973)

19 J, T. Schwartz, Editor, Mathematical aspects of computer science (New York City,

April 1966)

18 H. Grad, Editor, Magneto-fluid and plasma dynamics (New York City, April 1965)

17 R. Finn, Editor, Applications of nonlinear partial differential equations in mathematical

physics (New York City, April 1964)

16 R. Bel lman, Editor, Stochastic processes in mathematical physics and engineering (New

York City, April 1963)

15 N . C. Metropol is , A. H. Taub, J. Todd, and C. B . Tompkins, Editors, Experimental arithmetic, high speed computing, and mathematics (Atlantic City and Chicago, April 1962)

14 R. Bel lman, Editor, Mathematical problems in the biological sciences (New York City,

April 1961) 13 R. Bel lman, G. Birkhoff, and C. C. Lin, Editors, Hydrodynamic instability (New

York City, April 1960) 12 R. Jakobson, Editor, Structure of language and its mathematical aspects (New York

City, April 1960) 11 G. Birkhoff and E. P. Wigner , Editors, Nuclear reactor theory (New York City, April

1959) 10 R. Be l lman and M. Hall, Jr., Editors, Combinatorial analysis (New York University,

April 1957)

9 G. Birkhoff and R. E. Langer, Editors, Orbit theory (Columbia University, April

1958)

8 L. M. Graves, Editor, Calculus of variations and its applications (University of Chicago,

April 1956)

7 L. A. MacColl , Editor, Applied probability (Polytechnic Institute of Brooklyn, April

1955)

6 J. H. Curtiss , Editor, Numerical analysis (Santa Monica City College, August 1953)

5 A. E. Heins , Editor, Wave motion and vibration theory (Carnegie Institute of

Technology, June 1952)

4 M. H. Martin, Editor, Fluid dynamics (University of Maryland, June 1951)

3 R. V. Churchill, Editor, Elasticity (University of Michigan, June 1949)

2 A. H. Taub, Editor, Electromagnetic theory (Massachusetts Institute of Technology,

July 1948)

1 E. Reissner, Editor, Non-linear problems in mechanics of continua (Brown University, August 1947)