American Mathematical Society

Gerald Teschl

Graduate Studies in Mathematics

Volume 140

Ordinary DifferentialEquations andDynamical Systems

Ordinary Differential Equations and Dynamical Systems

Gerald Teschl

American Mathematical SocietyProvidence, Rhode Island

Graduate Studies in Mathematics

Volume 140

http://dx.doi.org/10.1090/gsm/140

EDITORIAL COMMITTEE

David Cox (Chair)Daniel S. FreedRafe Mazzeo

Gigliola Staffilani

2010 Mathematics Subject Classification. Primary 34-01, 37-01.

For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-140

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Teschl, Gerald, 1970–Ordinary differential equations and dynamical systems / Gerald Teschl.

p. cm. — (Graduate studies in mathematics ; v. 140)Includes bibliographical references and index.ISBN 978-0-8218-8328-0 (alk. paper)1. Differential equations—Textbooks. 2. Dynamics—Textbooks. I. Title.

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10 9 8 7 6 5 4 3 2 1 17 16 15 14 13 12

To Susanne, Simon, and Jakob

Contents

Preface ix

Part 1. Classical theory

Chapter 1. Introduction 3

§1.1. Newton’s equations 3

§1.2. Classification of differential equations 6

§1.3. First-order autonomous equations 9

§1.4. Finding explicit solutions 13

§1.5. Qualitative analysis of first-order equations 20

§1.6. Qualitative analysis of first-order periodic equations 28

Chapter 2. Initial value problems 33

§2.1. Fixed point theorems 33

§2.2. The basic existence and uniqueness result 36

§2.3. Some extensions 39

§2.4. Dependence on the initial condition 42

§2.5. Regular perturbation theory 48

§2.6. Extensibility of solutions 50

§2.7. Euler’s method and the Peano theorem 54

Chapter 3. Linear equations 59

§3.1. The matrix exponential 59

§3.2. Linear autonomous first-order systems 66

§3.3. Linear autonomous equations of order n 74

v

vi Contents

§3.4. General linear first-order systems 80

§3.5. Linear equations of order n 87

§3.6. Periodic linear systems 91

§3.7. Perturbed linear first-order systems 97

§3.8. Appendix: Jordan canonical form 103

Chapter 4. Differential equations in the complex domain 111

§4.1. The basic existence and uniqueness result 111

§4.2. The Frobenius method for second-order equations 116

§4.3. Linear systems with singularities 130

§4.4. The Frobenius method 134

Chapter 5. Boundary value problems 141

§5.1. Introduction 141

§5.2. Compact symmetric operators 146

§5.3. Sturm–Liouville equations 153

§5.4. Regular Sturm–Liouville problems 155

§5.5. Oscillation theory 166

§5.6. Periodic Sturm–Liouville equations 175

Part 2. Dynamical systems

Chapter 6. Dynamical systems 187

§6.1. Dynamical systems 187

§6.2. The flow of an autonomous equation 188

§6.3. Orbits and invariant sets 192

§6.4. The Poincare map 197

§6.5. Stability of fixed points 198

§6.6. Stability via Liapunov’s method 201

§6.7. Newton’s equation in one dimension 203

Chapter 7. Planar dynamical systems 209

§7.1. Examples from ecology 209

§7.2. Examples from electrical engineering 215

§7.3. The Poincare–Bendixson theorem 220

Chapter 8. Higher dimensional dynamical systems 229

§8.1. Attracting sets 229

§8.2. The Lorenz equation 234

Contents vii

§8.3. Hamiltonian mechanics 238

§8.4. Completely integrable Hamiltonian systems 243

§8.5. The Kepler problem 247

§8.6. The KAM theorem 250

Chapter 9. Local behavior near fixed points 255

§9.1. Stability of linear systems 255

§9.2. Stable and unstable manifolds 257

§9.3. The Hartman–Grobman theorem 264

§9.4. Appendix: Integral equations 270

Part 3. Chaos

Chapter 10. Discrete dynamical systems 281

§10.1. The logistic equation 281

§10.2. Fixed and periodic points 284

§10.3. Linear difference equations 287

§10.4. Local behavior near fixed points 288

Chapter 11. Discrete dynamical systems in one dimension 293

§11.1. Period doubling 293

§11.2. Sarkovskii’s theorem 296

§11.3. On the definition of chaos 297

§11.4. Cantor sets and the tent map 300

§11.5. Symbolic dynamics 303

§11.6. Strange attractors/repellers and fractal sets 309

§11.7. Homoclinic orbits as source for chaos 313

Chapter 12. Periodic solutions 317

§12.1. Stability of periodic solutions 317

§12.2. The Poincare map 319

§12.3. Stable and unstable manifolds 321

§12.4. Melnikov’s method for autonomous perturbations 324

§12.5. Melnikov’s method for nonautonomous perturbations 329

Chapter 13. Chaos in higher dimensional systems 333

§13.1. The Smale horseshoe 333

§13.2. The Smale–Birkhoff homoclinic theorem 335

§13.3. Melnikov’s method for homoclinic orbits 336

viii Contents

Bibliographical notes 341

Bibliography 345

Glossary of notation 349

Index 351

Preface

About

When you publish a textbook on such a classical subject the first ques-tion you will be faced with is: Why another book on this subject? Ev-erything started when I was supposed to give the basic course on OrdinaryDifferential Equations in Summer 2000. (At that time the course met 5 hoursper week.) While there were many good books on the subject available, noneof them quite fit my needs. I wanted a concise but rigorous introduction withfull proofs that also covered classical topics such as Sturm–Liouville bound-ary value problems, differential equations in the complex domain, as wellas modern aspects of the qualitative theory of differential equations. Thecourse was continued with a second part on Dynamical Systems and Chaosin Winter 2000/01, and the notes were extended accordingly. Since thenthe manuscript has been rewritten and improved several times according tothe feedback I got from students over the years when I redid the course.Moreover, since I had the notes on my homepage from the very beginning,this triggered a significant amount of feedback as well, from students whoreported typos, incorrectly phrased exercises, etc., to colleagues who re-ported errors in proofs and made suggestions for improvements, to editorswho approached me about publishing the notes. All this interest eventuallyresulted in a Chinese translation of an earlier version of the book. Moreover,if you google for the manuscript, you can see that it is used at several placesworldwide, linked as a reference at various sites, including Wikipedia. Fi-nally, Google Scholar will tell you that it is even cited in several publications.Hence I decided that it was time to turn it into a real book.

ix

x Preface

Content

This book’s main aim is to give a self-contained introduction to the fieldof ordinary differential equations with emphasis on the dynamical systemspoint of view while still keeping an eye on classical tools as pointed outbefore.

The first part is what I typically cover in the introductory course forbachelor’s level students. Of course it is typically not possible to cover ev-erything and one has to skip some of the more advanced sections. Moreover,it might also be necessary to add some material from the first chapter of thesecond part to meet curricular requirements.

The second part is a natural continuation beginning with planar exam-ples (culminating in the generalized Poincare–Bendixson theorem), continu-ing with the fact that things get much more complicated in three and moredimensions, and ending with the stable manifold and the Hartman–Grobmantheorem.

The third and last part gives a brief introduction to chaos, focusingon two selected topics: Interval maps with the logistic map as the primeexample plus the identification of homoclinic orbits as a source for chaosand the Melnikov method for perturbations of periodic orbits and for findinghomoclinic orbits.

Prerequisites

This book requires only some basic knowledge of calculus, complex func-tions, and linear algebra. In addition, I have tried to show how a computersystem, Mathematica1, can help with the investigation of differential equa-tions. However, the course is not tied to Mathematica and any similar pro-gram can be used as well.

Updates

The AMS is hosting a Web page for this book at

http://www.ams.org/bookpages/gsm-140/

where updates, corrections, and other material may be found, including alink to material on my website:

http://www.mat.univie.ac.at/~gerald/ftp/book-ode/

1Mathematica�

is a registered trademark of Wolfram Research, Inc.

Preface xi

There you can also find an accompanying Mathematica notebook with thecode from the text plus some additional material.

Acknowledgments

I wish to thank my students, Ada Akerman, Kerstin Ammann, JorgArnberger, Alexander Beigl, Paolo Capka, Jonathan Eckhardt, Michael Fis-cher, Anna Geyer, Ahmed Ghneim, Hannes Grimm-Strele, Tony Johansson,Klaus Kroncke, Alice Lakits, Simone Lederer, Oliver Leingang, JohannaMichor, Thomas Moser, Markus Muller, Andreas Nemeth, Andreas Pich-ler, Tobias Preinerstorfer, Jin Qian, Dominik Rasipanov, Martin Ringbauer,Reinaldo Garcia Rosario, Simon Roßler, Robert Stadler, Shelby Stanhope,Raphael Stuhlmeier, Gudrun Szewieczek, Gerhard Tulzer, Paul Wedrich,Florian Wisser, and colleagues, Edward Dunne, Klemens Fellner, GiuseppeFerrero, Ilse Fischer, Delbert Franz, Heinz Hanßmann, Daniel Lenz, JimSochacki, and Eric Wahlen, who have pointed out several typos and madeuseful suggestions for improvements. Finally, I would like to thank theanonymous referees for valuable suggestions that improved the presentationof the material.

If you find any errors or if you have comments or suggestions(no matter how small), please let me know.

I have been supported by the Austrian Science Fund (FWF) during muchof this writing, most recently under Grant Y330.

Gerald Teschl

Vienna, AustriaApril 2012

Gerald TeschlFakultat fur MathematikNordbergstraße 15Universitat Wien1090 Wien, Austria

E-mail: Gerald.Teschl@univie.ac.at

URL: http://www.mat.univie.ac.at/~gerald/

Bibliographical notes

The aim of this section is not to give a comprehensive guide to the literature,but to document the sources from which I have learned the materials andwhich I have used during the preparation of this text. In addition, I willpoint out some standard references for further reading.

Chapter 2: Initial value problems

The material in this section is, of course, standard. Classical referencesare Coddington and Levinson [7], Hartman [14], Hale [13], Ince [23], andWalter [44]. More modern introductions are Amann [2], Arnold [4], Hirsch,Smale, and Devaney [18], Robinson [36], Verhulst [43], and Wiggins [48].

Further uniqueness results can be found in the book by Walter [44] (seethe supplement to §12). There you can also find further technical improve-ments, in particular, for the case alluded to in the remark after Corollary 2.6(see the second supplement to §10).

More on Mathematica in general can be found in the standard documen-tation [49] and in connection with differential equations in [11] and [39].

General purpose references are the handbooks by Kamke [24] and Zwill-inger [50].

Chapter 3: Linear equations

Again this material is mostly standard and the same references as those forthe previous chapter apply. More information in particular on n’th orderequations can be found in Coddington and Levinson [7], Hartman [14], andInce [23].

341

342 Bibliographical notes

Chapter 4: Differential equations in the complex domain

Classical references with more information on this topic include Coddingtonand Levinson [7], Hille [17], and Ince [23]. For a more modern point ofview see Ilyashenko and Yakovenko [21]. The topics here are also closelyconnected with the theory of special functions. See Beals and Wong [5] fora modern introduction. For a collection of properties of special function astandard reference is the NIST Handbook of Mathematical Functions [30].

Chapter 5: Boundary value problems

Classical references include Coddington and Levinson [7] and Hartman [14].A nice informal treatment (although in German) can be found in Janich [22].More on Hill’s equation can be found in Magnus and Winkler [28]. For amodern introduction to singular Sturm–Liouville problems, see the booksby Weidmann [45], [46], my textbook [42], and the book by Levitan andSargsjan [27]. A reference with more applications and numerical methodsis by Hastings and McLeod [16].

Chapter 6: Dynamical systems

Classical references include Chicone [6], Guckenheimer and Holmes [12],Hasselblat and Katok [15], [25], Hirsch, Smale, and Devaney [18], Palis andde Melo [33], Perko [34], Robinson [35], [36], Ruelle [38], Verhulst [43], andWiggins [47], [48]. In particular, [15] and [25] emphasize ergodic theory,which is not covered here.

More on the connections with Lie groups and symmetries of differentialequations, briefly mentioned in Problem 6.5, can be found in the monographby Olver [31].

Chapter 7: Planar dynamical systems

The proof of the Poincare–Bendixson theorem follows Palis and de Melo[33]. More on ecological models can be found in Hofbauer and Sigmund[19]. Hirsch, Smale, and Devaney [18] and Robinson [36] also cover thesetopics nicely.

Chapter 8: Higher dimensional dynamical systems

More on the Lorenz equation can be found in the monograph by Sparrow[40]. The classical reference for Hamiltonian systems is, of course, Arnold’sbook [3] (see also [4]) as well as the monograph by Abraham, Marsden, andRatiu [1], which also contains extensions to infinite-dimensional systems.Other references are the notes by Moser [29] and the monograph by Wiggins[47]. A brief overview can be found in Verhulst [43].

Chapter 9: Local behavior near fixed points

The classical reference here is Hartman [14]. See also Coddington and Levin-son [7], Hale [13], Robinson [35], and Ruelle [38].

Bibliographical notes 343

Chapter 10: Discrete dynamical systems

One of the classical references is the book by Devaney [8]. A nice introduc-tion is provided by Holmgren [20]. Further references are Hasselblat andKatok [15], [25] and Robinson [36].

Chapter 11: Discrete dynamical systems in one dimension

The classical reference here is Devaney [8]. More on the Hausdorff measurecan be found in Falconer [9]. See also Holmgren [20] and Robinson [36].

Chapter 12: Periodic solutions

For more information see Chicone [6], Robinson [35], [36], and Wiggins [47].

Chapter 13: Chaos in higher dimensional systems

A proof of the Smale–Birkhoff theorem can be found in Robinson [35]. Seealso Chicone [6], Guckenheimer and Holmes [12], and Wiggins [47].

Bibliography

[1] R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis, andApplications, 2nd ed., Springer, New York, 1983.

[2] H. Amann, Ordinary Differential Equations, deGryter, Berlin, 1990.

[3] V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer,New York, 1989.

[4] V. I. Arnold, Ordinary Differential Equations: An Introduction to NonlinearAnalysis, Springer, Berlin, 1992.

[5] R. Beals and R. Wong, Special Functions, Cambridge University Press, Cam-bridge, 2010.

[6] C. Chicone, Ordinary Differential Equations with Applications, Springer, NewYork, 1999.

[7] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations,McGraw-Hill, New York, 1955.

[8] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley, Redwood City, 1989.

[9] K. Falconer, Fractal Geometry, Benjamin/Clummings Publishing, Menlo Park,1986.

[10] F. R. Gantmacher, Applications of the Theory of Matrices, Interscience, NewYork, 1959.

[11] A. Gray, M. Mezzino, and M. A. Pinsky, Introduction to Ordinary DifferentialEquations with Mathematica, Springer, New York, 1997.

[12] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems,and Bifurcations of Vector Fields, Springer, New York, 1983.

[13] J. Hale, Ordinary Differential Equations, Krieger, Malabar, 1980.

[14] P. Hartman, Ordinary Differential Equations, 2nd ed., SIAM, Philadelphia, 2002.

[15] B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge UP, Cam-bridge, 2003.

[16] S. P. Hastings and J. B. McLeod, Classical Methods in Ordinary DifferentialEquations: With Applications to Boundary Value Problems, Graduate Studies inMathematics, vol. 129, Amer. Math. Soc., Providence, RI, 2011.

345

346 Bibliography

[17] E. Hille, Ordinary Differential Equations in the Complex Domain, Dover, Mi-nolea,1997.

[18] M. W. Hirsch, S. Smale, and R. L. Devaney, Differential Equations, DynamicalSystems, and an Introduction to Chaos, Elsevier, Amsterdam, 2004.

[19] J. Hofbauer and K. Sigmund, Evolutionary Games and Replicator Dynamics,Cambridge University Press, Cambridge, 1998.

[20] R. A. Holmgren, A First Course in Discrete Dynamical Systems, 2nd ed.,Springer, New York, 1996.

[21] Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations,Graduate Studies in Mathematics, vol. 86, Amer. Math. Soc., Providence, RI,2008.

[22] K. Janich, Analysis, 2nd ed., Springer, Berlin, 1990.

[23] E. L. Ince, Ordinary Differential Equations, Dover, New York, 1956.

[24] E. Kamke, Differentialgleichungen, I. Gewohnliche Differentialgleichungen,Springer, New York, 1997.

[25] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of DynamicalSystems, Cambridge UP, Cambridge, 1995.

[26] J. L. Kelly, General Topology, Springer, New York, 1955.

[27] B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory: SelfadjointOrdinary Differential Operators, Translations of Mathematical Monographs, vol.39, Amer. Math. Soc., Providence, RI, 1975.

[28] W. Magnus and S. Winkler, Hill’s Equation, Dover, Minolea, 2004.

[29] J. Moser, Stable and Random Motions in Dynamical Systems: With Special Em-phasis on Celestial Mechanics, Princeton University Press, Princeton, 2001.

[30] F. W. J. Olver et al., NIST Handbook of Mathematical Functions, CambridgeUniversity Press, Cambridge, 2010.

[31] P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed.,Springer, New York, 1993.

[32] R. S. Palais, The symmetries of solitons, Bull. Amer. Math. Soc. 34 (1997),339–403.

[33] J. Palis and W. de Melo, Geometric Theory of Dynamical Systems, Springer,New York, 1982.

[34] L. Perko, Differential Equations and Dynamical Systems, 2nd ed., Springer, NewYork, 1996.

[35] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,CRC Press, Boca Raton, 1995.

[36] C. Robinson, Introduction to Dynamical Systems: Discrete and Continuous,Prentice Hall, New York, 2004.

[37] C. A. Rogers, Hausdorff Measures, Cambridge University Press, Cambridge,1970.

[38] D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory, Aca-demic Press, San Diego, 1988.

[39] D. Schwalbe and S. Wagon, VisualDSolve. Visualizing Differential Equationswith Mathematica, Springer, New York, 1997.

[40] C. Sparrow, The Lorenz Equation, Bifurcations, Chaos and Strange Attractors,Springer, New York, 1982.

Bibliography 347

[41] E. Stein and R. Shakarchi, Complex Analysis, Princeton UP, Princeton, 2003.

[42] G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications toSchrodinger Operators, Graduate Studies in Mathematics, vol. 99, Amer. Math.Soc., Providence, RI, 2009.

[43] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, 2nd ed.,Springer, Berlin, 2000.

[44] W. Walter, Ordinary Differential Equations, Springer, New York, 1998.

[45] J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York, 1980.

[46] J. Weidmann, Spectral Theory of Ordinary Differential Operators, Lecture Notesin Mathematics, vol. 1258, Springer, Berlin, 1987.

[47] S. Wiggins, Global Bifurcations and Chaos, 2nd ed., Springer, New York, 1988.

[48] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,2nd ed., Springer, New York, 2003.

[49] S. Wolfram, The Mathematica Book, 4th ed., Wolfram Media/Cambridge Uni-versity Press, Champaign/Cambridge, 1999.

[50] D. Zwillinger, Handbook of Differential Equations, 3rd ed., Academic Press, SanDiego, 1997.

Glossary of notation

A± . . . matrix A restricted to E±(A), 264Br(x) . . . open ball of radius r centered at xC(U, V ) . . . set of continuous functions from U to VCb(U, V ) . . . set of bounded continuous functions from U to VC(U) = C(U,R)Ck(U, V ) . . . set of k times continuously differentiable functionsC . . . the set of complex numbersχA . . . Characteristic polynomial of A, 103d(U) . . . diameter of U , 309d(x, y) . . . distance in a metric spaced(x,A) . . . distance between a point x and a set A, 196

dfx = ∂f∂x Jacobian matrix of a differentiable mapping f at x

δj,k . . . Kronecker delta: δj,j = 1 and δj,k = 0 if j �= kE0(A) . . . center subspace of a matrix, 109E±(A) . . . (un)stable subspace of a matrix, 109Fix(f) = {x|f(x) = x} set of fixed points of f , 284γ(x) . . . orbit of x, 192γ±(x) . . . forward, backward orbit of x, 192Γ(z) . . . Gamma function, 126H0 . . . inner product space, 146I . . . identity matrixIx = (T−(x), T+(x)) maximal interval of existence, 189Ker(A) . . . kernel of a matrixLμ . . . logistic map, 282Λ . . . a compact invariant setM± . . . (un)stable manifold, 258, 322

349

350 Glossary of notation

N = {1, 2, 3, . . . } the set of positive integersN0 = N ∪ {0}o(.) . . . Landau symbolO(.) . . . Landau symbolΩ(f) . . . set of nonwandering points, 196PΣ(y) . . . Poincare map, 198Per(f) = {x|f(x) = x} set of periodic points of f , 284Φ(t, x0) . . . flow of a dynamical system, 189Π(t, t0) . . . principal matrix of a linear system, 81R . . . the set of realsRan(A) . . . range of a matrixσ . . . shift map on ΣN , 305σ(A) . . . spectrum (set of eigenvalues) of a matrix, 103ΣN . . . sequence space over N symbols, 304sign(x) . . . +1 for x > 0 and −1 for x < 0; sign functionT±(x) . . . positive, negative lifetime of x, 192T (x) . . . period of x (if x is periodic), 192Tμ . . . tent map, 299ω±(x) . . . positive, negative ω-limit set of x, 193W± . . . (un)stable set, 257, 231, 284Z = {. . . ,−2,−1, 0, 1, 2, . . . } the set of integersz . . . a complex number√z . . . square root of z with branch cut along (−∞, 0)

z∗ . . . complex conjugation‖.‖ . . . norm in a Banach space|.| . . . Euclidean norm in R

n respectively Cn

〈., ..〉 . . . scalar product in H0, 146(λ1, λ2) = {λ ∈ R |λ1 < λ < λ2}, open interval[λ1, λ2] = {λ ∈ R |λ1 ≤ λ ≤ λ2}, closed interval�x = max{n ∈ Z|n ≤ x}, floor function�x� = min{n ∈ Z|n ≥ x}, ceiling functiona ∧ b = cross product in R

3

Index

Abel’s identity, 83

action integral, 238

action variable, 244

adjoint matrix, 103

analytic, 111

angle variable, 245

angular momentum, 242, 248arc, 220

Arzela–Ascoli theorem, 55

asymptotic phase, 323

asymptotic stability, 71, 198, 286, 317

attracting set, 231

attractor, 233, 309

strange, 309

autonomous differential equation, 7

backward asymptotic, 285

Banach algebra, 66

Banach space, 34

basin of attraction, 231

basis

orthonormal, 149

Bendixson criterion, 227Bernoulli equation, 15

Bessel

equation, 122

function, 123

inequality, 148

bifurcation, 21

diagram, 295

pitchfork, 200

Poincare–Andronov–Hopf, 220

point, 294

saddle-node, 200

theory, 200

transcritical, 200

boundary condition, 144, 156

antiperiodic, 177

Dirichlet, 156

Neumann, 156periodic, 177

Robin, 156

boundary value problem, 144

canonical transform, 243

Cantor set, 301

Caratheodory, 42

catenary, 19

Cauchy sequence, 33

Cauchy–Hadamard theorem, 112

Cauchy–Schwarz inequality, 147

center, 69

characteristic

exponents, 93, 118, 138

multipliers, 93

characteristic polynomial, 103commutator, 61

competitive system, 213

complete, 34

completely integrable, 245

confluent hypergeometric equation, 128

conjugacy

topological, 268

constant of motion, 202, 240

contraction, 35

contraction principle, 35

351

352 Index

cooperative system, 213cover, 309cyclic vector, 106

d’Alembert reduction, 84, 88d’Alembert’s formula, 145damping

critical, 78over, 78under, 78

damping factor, 79diameter, 309difference equation, 126, 283differential equation

autonomous, 7exact, 18homogeneous, 7, 15hyperbolic, 256integrating factor, 18linear, 7order, 6ordinary, 6partial, 7separable, 11solution, 6system, 7

diophantine condition, 251directional field, 16Dirichlet boundary condition, 156domain of attraction, 231dominating function, 272Duffing equation, 233, 263, 339Duhamel’s formula, 72Dulac criterion, 227dynamical system, 187

chaotic, 298continuous, 188discrete, 187invertible, 187

eigenfunction, see eigenvectoreigenspace, 103, 149

generalized, 105eigenvalue, 103, 149

simple, 149eigenvector, 103, 149eigenvectors

generalized, 105Einstein equation, 242entire function, 153equicontinuous, 55equilibrium point, see fixed point

equivalencetopological, 298

error function, 89Euler equation, 18, 116Euler system, 131Euler’s formula, 67Euler’s reflection formula, 127Euler–Lagrange equations, 239Euler–Mascheroni constant, 124exponential stability, 199

Fermi–Pasta–Ulam experiment, 247Fibonacci numbers, 288first integral, 240first variational equation, 46

periodic, 318fixed point, 35, 192, 284

asymptotically stable, 198, 286exponentially stable, 199hyperbolic, 257stable, 198unstable, 198

fixed-point theoremcontraction principle, 35Weissinger, 39

Floquetdiscriminant, 176exponents, 93multipliers, 93, 176solutions, 176

flow, 189forcing, 79forward asymptotic, 284Fourier cosine series, 165Fourier sine series, 143, 164, 165Frobenius method, 138from domain, 161Fuchs system, 138fundamental matrix solution, 83

Gamma function, 126Gauss error function, 89geodesics, 241global solution, 51gradient systems, 203Green function, 158Grobman–Hartman theorem, 266Gronwall inequality, 42group, 187

Hamilton mechanics, 206, 239Hamilton principle, 238

Index 353

Hammerstein integral equation, 275Hankel function, 125harmonic numbers, 124harmonic oscillator, 246Hartman–Grobman theorem, 266

maps, 288Hausdorff dimension, 311Hausdorff measure, 310heat equation, 145Heun’s method, 57Hilbert space, 146Hilbert’s 16th problem, 226Hilbert–Schmidt operator, 164Hill equation, 93homoclinic orbit, 315homoclinic point, 315, 335

transverse, 335homoclinic tangle, 336Hopf bifurcation, 220, 324Hurwitz matrix, 71hyperbolic, 256, 257hypergeometric equation, 128

indicial equation, 118inequality

Cauchy–Schwarz, 147Gronwall, 42

initial value problem, 36inner product, 146

space, 146integral curve, 189

maximal, 189integral equation, 36

Hammerstein, 275Volterra, 273

integrating factor, 18invariant

set, 193, 284subspace, 103

isoclines, 24itinerary map, 302, 313, 314

Jacobi identity, 242Jacobian matrix, 39Jordan block, 106Jordan canonical form, 61, 107

real, 65Jordan curve, 220

Kepler’s laws for planetary motion, 249Kirchhoff’s laws, 76Korteweg–de Vries equation, 207

Krasovskii–LaSalle principle, 202Kronecker torus, 251Kummer function, 128

Lagrange function, 238Lagrange identity, 157Laplace transform, 73LaSalle principle, 202Laurent series, 116Lax equation, 247Lax pair, 247Legendre equation, 128Legendre transform, 239Leibniz’ rule, 242Lienard equation, 216Liapunov function, 201, 286

strict, 201, 286Liapunov–Schmidt reduction, 330Lie derivative, 202Lie group, 191Lie series, 192lifetime, 192limit cycle, 226Liouville’s formula, 83, 237Lipschitz continuous, 27, 37logistic map, 282Lorenz equation, 234Lotka–Volterra equation, 209lower solution, 24

manifold(un)stable, fixed point, 258, 289(un)stable, linear, 255(un)stable, periodic point, 323center, linear, 255stable, 289unstable, 289

mass spectrometry, 96mathematical pendulum, 204Mathieu equation, 95matrix

adjoint, 103exponential, 60Hurwitz, 71logarithm, 108norm, 60orthogonal, 104symmetric, 104symplectic, 240unitary, 104

maximal solution, 51measure

354 Index

Hausdorff, 310outer, 310

Melnikov integralhomoclinic, 339periodic, 326

minimal polynomial, 105monodromy matrix, 91, 176movable singularity, 131

N -body problem, 250Neumann boundary condition, 156Neumann series, 270Newton’s second law of motion, 3nilpotent, 106nonresonant, 251nonwandering, 196, 286norm, 33

matrix, 60operator, 270

normalized, 146

Ohm’s law, 77omega limit set, 193, 229one-parameter Lie group, 191operator

bounded, 150compact, 150domain, 149linear, 149symmetric, 149

orbit, 192, 284asymptotically stable, 317closed, 192heteroclinic, 262, 291homoclinic, 262, 291periodic, 192, 284stable, 317

ordereigenvector, 105

orthogonal, 146orthogonal matrix, 104orthonormal basis, 104oscillating, 173Osgood uniqueness criterion, 58

Painleve transcendents, 131parallel, 146parallelogram law, 152Peano theorem, 56pendulum, 204perfect, 301period anulus, 326

isochronous, 331regular, 331

period doubling, 295periodic orbit, 192, 284

stable, 286periodic point, 192, 284

attracting, 285hyperbolic, 286period, 192repelling, 285

periodic solutionstability, 317

perpendicular, 146phase space, 203Picard iteration, 38Picard–Lindelof theorem, 38pitchfork bifurcation, 200Pochhammer symbol, 123Poincare map, 29, 198, 319Poincare–Andronov–Hopf bifurcation,

220point

fixed, 192nonwandering, 196, 286recurrent, 286

Poisson bracket, 240power series, 112principal matrix solution, 82projection, 109, 110Prufer variables, 166

modified, 172Pythagorean theorem, 147

quadratic form, 161quadrupole mass spectrometry, 96quasi-periodic, 251

radius of convergence, 112Rayleigh–Ritz principle, 162recurrent, 286reduction of order, 84, 88regular perturbation, 48regular point, 192relativistic mechanics, 242repeller, 309

strange, 309resolvent, 158resonance catastrophy, 79resonance frequency, 79resonant, 251Riccati equation, 15, 90, 154Riemann equation, 129

Index 355

Riemann symbol, 129RLC circuit, 77Robin boundary condition, 156Rofe-Beketov formula, 154Routh–Hurwitz criterion, 71Runge–Kutta algorithm, 57

saddle, 68saddle-node bifurcation, 200Sarkovskii ordering, 297scalar product, 104, 146Schrodinger equation, 86semigroup, 187sensitive dependence, 297separation of variables, 142sesquilinear form, 146set

attracting, 231, 309hyperbolic attracting, 309hyperbolic repelling, 309invariant, 193, 284repelling, 309

shift map, 302, 305singular point, see fixed pointsingularity

movable, 131regular, 133simple, 133weak, 133

sink, 68Smale horseshoe, 333small divisor, 251snapback repeller, 315soliton, 207solution

lower, 24matrix, 83, 287sub, 24super, 24upper, 24

source, 68spectral radius, 110spectral theorem, 151spectrum, 103stability, 71, 198, 286, 317stable set, 231, 257, 285stationary point, see fixed pointstrange attractor, 237Sturm–Liouville problem, 144sub solution, 24submanifold, 197subshift of finite type, 306

subspacecenter, 109invariant, 103reducing, 103stable, 109unstable, 109

superposition principle, 81symbol space, 304symmetric matrix, 104symplectic

gradient, 240group, 243map, 243matrix, 240two form, 243

tent map, 299theorem

Arzela–Ascoli, 55, 159Cauchy–Hadamard, 112Cayley–Hamilton, 107dominated convergence, 272Floquet, 92Fuchs, 119, 121Hartman–Grobman, 266, 288Jordan curve, 220KAM, 251Kneser, 174Krasovskii–LaSalle, 202Liapunov, 202Melnikov, 339Noether, 240Osgood, 58Peano, 56Picard–Lindelof, 38

improved, 40Poincare’s recurrence, 241Poincare–Bendixson, 222, 223Pythagorean, 147Routh–Hurwitz, 71Smale–Birkhoff homoclinic, 336stable manifold, 261, 290, 322Sturm’s comparison, 170uniform contraction principle, 270Weissinger, 39

time-one map, 237totally disconnected, 301, 304trajectory, 189transcritical bifurcation, 200transformation

fiber preserving, 14transition matrix, 306

356 Index

transitive, 306transitive, 233, 298trapping region, 232traveling wave ansatz, 207triangle inequality, 33

inverse, 33two-body problem, 247

uniform contraction principle, 270unit vector, 146unitary matrix, 104unstable, 198unstable set, 231, 257, 285upper solution, 24

van der Pol equation, 219variable

dependent, 6independent, 6

variation of constants (parameters), 84vector field, 188

complete, 193vector space, 33

inner product space, 146normed, 33

Volterra integral equation, 273Volterra–Lotka equation, 209

wave equation, 141Weierstraß elliptic function, 207well-posed, 42Weyl asymptotics, 173Weyl–Titchmarsh m-functions, 176Wronski determinant, 83, 88Wronskian, 88

modified, 154

zeta function, 308Zorn’s lemma, 51

GSM/140

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This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students.

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