sign in sign up
<<
Home , Differential equation, Linear system

Dynamical Systems and Linear

Fritz Colonius Wolfgang Kliemann

Graduate Studies in Volume 158

American Mathematical Society Dynamical Systems and

https://doi.org/10.1090//gsm/158

Dynamical Systems and Linear Algebra

Fritz Colonius Wolfgang Kliemann

Graduate Studies in Mathematics Volume 158

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani

2010 Mathematics Subject Classification. Primary 15-01, 34-01, 37-01, 39-01, 60-01, 93-01.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-158

Library of Congress Cataloging-in-Publication Data Colonius, Fritz. Dynamical systems and linear algebra / Fritz Colonius, Wolfgang Kliemann. pages cm. – (Graduate studies in mathematics ; volume 158) Includes bibliographical references and index. ISBN 978-0-8218-8319-8 (alk. paper) 1. , Linear. 2. Topological dynamics. I. Kliemann, Wolfgang. II. Title.

QA184.2.C65 2014 512.5–dc23 2014020316

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2014 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 http://www.ams.org/ 10987654321 191817161514 This book is dedicated to the Institut f¨ur Dynamische Systeme at Universit¨at Bremen, which had a lasting influence on our mathematical thinking, as well as to our students. This book would not have been possible without the interaction at the Institut and the graduate programs in our departments.

Contents

Introduction xi

Notation xv

Part 1. Matrices and Linear Dynamical Systems

Chapter 1. Autonomous Linear Differential and Difference 3 §1.1. Existence of Solutions 3 §1.2. The Real Jordan Form 6 §1.3. Solution Formulas 10 §1.4. Lyapunov Exponents 12 §1.5. The Discrete-Time Case: Linear Difference Equations 18 §1.6. Exercises 24 §1.7. Orientation, Notes and References 27

Chapter 2. Linear Dynamical Systems in Rd 29 §2.1. Continuous-Time Dynamical Systems or Flows 29 §2.2. Conjugacy of Linear Flows 33 §2.3. Linear Dynamical Systems in Discrete Time 38 §2.4. Exercises 43 §2.5. Orientation, Notes and References 43

Chapter 3. Chain Transitivity for Dynamical Systems 47 §3.1. Sets and Chain Transitivity 47 §3.2. The Chain Recurrent Set 54

vii viii Contents

§3.3. The Discrete-Time Case 59 §3.4. Exercises 63 §3.5. Orientation, Notes and References 65 Chapter 4. Linear Systems in Projective Space 67 §4.1. Linear Flows Induced in Projective Space 67 §4.2. Linear Difference Equations in Projective Space 75 §4.3. Exercises 78 §4.4. Orientation, Notes and References 78 Chapter 5. Linear Systems on Grassmannians 81 §5.1. Some Notions and Results from 82 §5.2. Linear Systems on Grassmannians and Volume Growth 86 §5.3. Exercises 94 §5.4. Orientation, Notes and References 95

Part 2. Time-Varying Matrices and Linear Skew Product Systems Chapter 6. Lyapunov Exponents and Linear Skew Product Systems 99 §6.1. Existence of Solutions and Continuous Dependence 100 §6.2. Lyapunov Exponents 106 §6.3. Linear Skew Product Flows 113 §6.4. The Discrete-Time Case 118 §6.5. Exercises 121 §6.6. Orientation, Notes and References 123 Chapter 7. Periodic Linear Differential and Difference Equations 127 §7.1. Floquet Theory for Linear Difference Equations 128 §7.2. Floquet Theory for Linear Differential Equations 136 §7.3. The Mathieu 144 §7.4. Exercises 151 §7.5. Orientation, Notes and References 153 Chapter 8. Morse Decompositions of Dynamical Systems 155 §8.1. Morse Decompositions 155 §8.2. 159 §8.3. Morse Decompositions, Attractors, and Chain Transitivity 164 §8.4. Exercises 166 Contents ix

§8.5. Orientation, Notes and References 167 Chapter 9. Topological Linear Flows 169 §9.1. The Spectral Decomposition Theorem 170 §9.2. Selgrade’s Theorem 178 §9.3. The Morse Spectrum 184 §9.4. Lyapunov Exponents and the Morse Spectrum 192 §9.5. Application to Robust Linear Systems and Bilinear Control Systems 197 §9.6. Exercises 207 §9.7. Orientation, Notes and References 208 Chapter 10. Tools from Ergodic Theory 211 §10.1. Measures 211 §10.2. Birkhoff’s Ergodic Theorem 214 §10.3. Kingman’s Subadditive Ergodic Theorem 217 §10.4. Exercises 220 §10.5. Orientation, Notes and References 221 Chapter 11. Random Linear Dynamical Systems 223 §11.1. The Multiplicative Ergodic Theorem (MET) 224 §11.2. Some Background on Projections 233 §11.3. Singular Values, Exterior Powers, and the Goldsheid-Margulis Metric 237 §11.4. The Deterministic Multiplicative Ergodic Theorem 242 §11.5. The Furstenberg-Kesten Theorem and Proof of the MET in Discrete Time 252 §11.6. The Random Linear Oscillator 263 §11.7. Exercises 266 §11.8. Orientation, Notes and References 268 Bibliography 271 Index 279

Introduction

Background Linear algebra plays a key role in the theory of dynamical systems, and concepts from dynamical systems allow the study, characterization and gen- eralization of many objects in linear algebra, such as similarity of matrices, eigenvalues, and (generalized) eigenspaces. The most basic form of this in- terplay can be seen as a quadratic A gives rise to a discrete time xk+1 = Axk,k=0, 1, 2,... and to a continuous time dynamical system via the linear ordinary differential equationx ˙ = Ax. The (real) Jordan form of the matrix A allows us to write the solution of the differential equationx ˙ = Ax explicitly in terms of the matrix ex- ponential, and hence the properties of the solutions are intimately related to the properties of the matrix A. Vice versa, one can consider properties of a linear flow in Rd and infer characteristics of the underlying matrix A. Going one step further, matrices also define (nonlinear) systems on smooth manifolds, such as the sphere Sd−1 in Rd, the Grassmannian manifolds, the flag manifolds, or on classical (matrix) Lie groups. Again, the behavior of such systems is closely related to matrices and their properties. Since A.M. Lyapunov’s thesis [97] in 1892 it has been an intriguing prob- lem how to construct an appropriate linear algebra for time-varying systems. Note that, e.g., for stability of the solutions ofx ˙ = A(t)x it is not sufficient that for all t ∈ R the matrices A(t) have only eigenvalues with negative real part (see, e.g., Hahn [61], Chapter 62). Classical Floquet theory (see Floquet’s 1883 paper [50]) gives an elegant solution for the periodic case, but it is not immediately clear how to build a linear algebra around Lya- punov’s ‘order numbers’ (now called Lyapunov exponents) for more general time dependencies. The key idea here is to write the time dependency as a

xi xii Introduction dynamical system with certain recurrence properties. In this way, the mul- tiplicative ergodic theorem of Oseledets from 1968 [109] resolves the basic issues for measurable linear systems with stationary time dependencies, and the Morse spectrum together with Selgrade’s theorem [124] goes a long way in describing the situation for continuous linear systems with chain transitive time dependencies. A third important area of interplay between dynamics and linear algebra arises in the of nonlinear systems about fixed points or arbitrary . Linearization of a differential equationy ˙ = f(y)inRd about a d fixed point y0 ∈ R results in the linear differential equationx ˙ = f (y0)x and theorems of the type Grobman-Hartman (see, e.g., Bronstein and Kopan- skii [21]) resolve the behavior of the flow of the nonlinear equation locally around y0 up to conjugacy, with similar results for dynamical systems over a or chain recurrent base. These observations have important applications in the natural sciences and in design and analysis of systems. Specifically, they are the basis for stochastic (see, e.g., Arnold [6]), and robust stability and stabilizability (see, e.g., Colonius and Kliemann [29]). Stabil- ity radii (see, e.g., Hinrichsen and Pritchard [68]) describe the amount of perturbation the operating point of a system can sustain while remaining stable, and stochastic stability characterizes the limits of acceptable noise in a system, e.g., an electric power system with a substantial component of wind or wave based generation. Goal This book provides an introduction to the interplay between linear alge- bra and dynamical systems in continuous time and in discrete time. There are a number of other books emphasizing these relations. In particular, we would like to mention the book [69] by M.W. Hirsch and S. Smale, which always has been a great source of inspiration for us. However, this book restricts attention to autonomous equations. The same is true for other books like M. Golubitsky and M. Dellnitz [54]orF.Lowenthal[96], which is designed to serve as a text for a first course in linear algebra, and the relations to linear autonomous differential equations are established on an elementary level only. Our goal is to review the autonomous case for one d × d matrix A via induced dynamical systems in Rd and on Grassmannians, and to present the main nonautonomous approaches for which the time dependency A(t) is given via skew-product flows using periodicity, or topological (chain re- currence) or ergodic properties (invariant measures). We develop general- izations of (real parts of) eigenvalues and eigenspaces as a starting point Introduction xiii for a linear algebra for classes of time-varying linear systems, namely peri- odic, random, and perturbed (or controlled) systems. Several examples of (low-dimensional) systems that play a role in engineering and science are presented throughout the text. Originally, we had also planned to include some basic concepts for the study of genuinely nonlinear systems via linearization, emphasizing invari- ant manifolds and Grobman-Hartman type results that compare nonlinear behavior locally to the behavior of associated linear systems. We decided to skip this discussion, since it would increase the length of this book consider- ably and, more importantly, there are excellent treatises of these problems available in the literature, e.g., Robinson [117] for linearization at fixed points, or the work of Bronstein and Kopanskii [21] for more general lin- earized systems. Another omission is the rich interplay with the theory of Lie groups and semigroups where many concepts have natural counterparts. The mono- graph [48] by R. Feres provides an excellent introduction. We also do not treat nonautonomous differential equations via pullback or other fiberwise constructions; see, e.g., Crauel and Flandoli [37], Schmalfuß [123], and Ras- mussen [116]; our emphasis is on the treatment of families of nonautonomous equations. Further references are given at the end of the chapters. Finally, it should be mentioned that all concepts and results in this book can be formulated in continuous and in discrete time. However, sometimes results in discrete time may be easier to state and to prove than their ana- logues in continuous time, or vice versa. At times, we have taken the liberty to pick one convenient setting, if the ideas of a result and its proof are par- ticularly intuitive in the corresponding setup. For example, the results in Chapter 5 on induced systems on Grassmannians are formulated and derived only in continuous time. More importantly, the proof of the multiplicative ergodic theorem in Chapter 11 is given only in discrete time (the formula- tion and some discussion are also given in continuous time). In contrast, Selgrade’s Theorem for topological linear dynamical systems in Chapter 9 and the results on Morse decompositions in Chapter 8, which are used for its proof, are given only in continuous time. Our aim when writing this text was to make ‘time-varying linear alge- bra’ in its periodic, topological and ergodic contexts available to beginning graduate students by providing complete proofs of the major results in at least one typical situation. In particular, the results on the Morse spectrum in Chapter 9 and on multiplicative ergodic theory in Chapter 11 have de- tailed proofs that, to the best of our knowledge, do not exist in the current literature. xiv Introduction

Prerequisites The reader should have basic knowledge of (including met- ric spaces) and linear algebra. No previous exposure to ordinary differential equations is assumed, although a first course in linear differential equations certainly is helpful. Multilinear algebra shows up in two places: in Section 5.2 we discuss how the volumes of parallelepipeds grow under the flow of a linear autonomous differential equation, which we relate to chain recur- rent sets of the induced flows on Grassmannians. The necessary elements of multilinear algebra are presented in Section 5.1. In Chapter 11 the proof of the multiplicative ergodic theorem requires further elements of multilinear algebra which are provided in Section 11.3. Understanding the proofs in Chapter 10 on ergodic theory and Chapter 11 on random linear dynamical systems also requires basic knowledge of σ-algebras and probability mea- sures (actually, a detailed knowledge of Lebesgue should suffice). The results and methods in the rest of the book are independent of these additional prerequisites. Acknowledgements The idea for this book grew out of the preparations for Chapter 79 in the Handbook of Linear Algebra [71]. Then WK gave a course “Dynamics and Linear Algebra” at the Simposio 2007 of the Sociedad de Matem´atica de Chile. FC later taught a course on the same topic at Iowa State University within the 2008 Summer Program for Graduate Students of the Institute of Mathematics and Its Applications, Minneapolis. Parts of the manuscript were also used for courses at the University of Augsburg in the summer semesters 2010 and 2013 and at Iowa State University in Spring 2011. We gratefully acknowledge these opportunities to develop our thoughts, the feed- back from the audiences, and the financial support. Thanks for the preparation of figures are due to: Isabell Graf (Section 4.1), Patrick Roocks (Section 5.2); Florian Ecker and Julia Rudolph (Section 7.3.); and Humberto Verdejo (Section 11.6). Thanks are also due to Philipp D¨uren, Julian Braun, and Justin Peters. We are particularly indebted to Christoph Kawan who has read the whole manuscript and provided us with long lists of errors and inaccuracies. Special thanks go to Ina Mette of the AMS for her interest in this project and her continuous support during the last few years, even when the text moved forward very slowly. The authors welcome any comments, suggestions, or corrections you may have. Fritz Colonius Wolfgang Kliemann Institut f¨ur Mathematik Department of Mathematics Universit¨at Augsburg Iowa State University Notation

Throughout this text we will use the following notation:

A  B =(A \ B) ∪ (B \ A) , the symmetric difference of sets f −1(E)={x | f(x) ∈ E} for a map f : X → Y and E ⊂ Y Ec the complement of a subset E ⊂ X, Ec = X \ E IE the characteristic of a set E, IE(x):=1 if x ∈ E and IE(x) := 0 elsewhere f +(x)=max(f(x), 0), the positive part of f : X → R log+ x log+ x := log x for x ≥ 1 and log+ x := 0 for x ≤ 1 gl(d, R),gl(d, C) the set of real (complex) d × d matrices Gl(d, R),Gl(d, C) the set of invertible real (complex) d × d matrices A the transpose of a matrix A ∈ gl(d, R) · anormonRd or an induced matrix norm spec(A) the set of eigenvalues μ ∈ C of a matrix A imA, trA the image and the trace of a A,resp. lim sup, lim inf limit superior, limit inferior N N , 0 the√ set of natural numbers excluding and including 0 ıı= −1 z the complex conjugate of z ∈ C A¯ A¯ =(aij)forA =(aij) ∈ gl(d, C) Pd−1 the real projective space Pd−1 = RPd−1 d Gk(d)thekth Grassmannian of R L(λ) the Lyapunov space associated with a Lyapunov exponent λ E expectation (relative to a probability measure P )

For points x and nonvoid subsets E of a metric space X with metric d:

N(x, ε)={y ∈ X | d(x, y) <ε}, the ε-neighborhood of x diam E =sup{d(x, y) | x, y ∈ E}, the diameter of E dist(x, E) = inf {d(x, y) | y ∈ E} , the distance of x to E cl E,int E the topological closure and interior of E,resp.

xv

Bibliography

[1] D. Acheson, From to Chaos. An Introduction to Chaos, Oxford University Press 1997. [2] J.M. Alongi and G.S. Nelson, Recurrence and , Amer. Math. Soc., 2007. [3] E. Akin, The of Dynamical Systems, Amer. Math. Soc., 1993. [4] H. Amann, Ordinary Differential Equations, Walter de Gruyter, 1990. [5] H. Amann, J. Escher, Analysis. II, Birkh¨auser Verlag, Basel, 2008. [6] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. [7] L. Arnold, W. Kliemann, and E. Oeljeklaus, Lyapunov exponents of linear stochastic systems, Springer Lecture Notes in Mathematics No. 1186 (1986), 85-125. [8] L. Arnold and M. Scheutzow, Perfect cocycles through stochastic differential equations, Probab. Theory Relat. Fields 101 (1995), 65-88. [9] D.K. Arrowsmith and C.M. Place, An Introduction to Dynamical Systems,Cam- bridge University Press, 1990. [10] Z. Artstein and I. Bright, Periodic optimization suffices for infinite horizon planar , SIAM J. Control Optim. 48 (2010), 4963-4986. [11] J. Ayala-Hoffmann, P. Corbin, K. McConville, F. Colonius, W. Kliemann, and J. Peters, Morse decompositions, attractors and chain recurrence,Proyec- ciones Journal of Mathematics 25 (2006), 79-109. [12] V. Ayala, F. Colonius, and W. Kliemann, Dynamic characterization of the Lyapunov form of matrices, Linear Algebra and Its Applications 402 (2005), 272- 290. [13] V. Ayala, F. Colonius, and W. Kliemann, On topological equivalence of linear flows with applications to bilinear control systems, J. Dynamical and Control Systems 17 (2007), 337-362. [14] V. Ayala and C. Kawan, Topological conjugacy of real projective flows, J. London Math. Soc. 90 (2014), 49-66. [15] L. Barreira and Ya.B. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, University Lecture , vol. 23, Amer. Math. Soc., 2002.

271 272 Bibliography

[16] L. Barreira and Y.B. Pesin, Nonuniform Hyperbolicity, Cambridge University Press, 2007. [17] H. Baumgartel,¨ Analytic for Matrices and Operators, Birkh¨auser, 1985. [18] M.V. Bebutov, Dynamical systems in the space of continuous functions, Dokl. Akad. Nauk SSSR 27 (1940), 904-906 [19] W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Ge- ometry, Academic Press, New York, 1975. [20] C.J. Braga Barros and L.A.B. San Martin, Chain transitive sets for flows on flag bundles. Forum Mathematicum 19 (2007), 19-60. [21] I.U. Bronstein and A.Ya Kopanskii, Smooth Invariant Manifolds and Normal Forms, World Scientific, 1994. [22] A. Bruckner, J. Bruckner, and B. Thompson, Real Analysis, Prentice Hall, 1997. [23] B. F. Bylov, R. E. Vinograd, D. M. Grobman, and V. V. Nemytskii, Theory of Lyapunov Exponents, Nauka, 1966 (in Russian). [24] L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer-Verlag, 1971, Third edition. [25] F. Chatelin, Eigenvalues of Matrices, SIAM 2012, Revised edition. [26] C. Chicone, Ordinary Differential Equations with Applications, Springer-Verlag, 1999. [27] C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs, 70, American Mathematical Society, Providence, RI, 1999. [28] G.H. Choe, Computational Ergodic Theory, Springer-Verlag, 2005. [29] F. Colonius and W. Kliemann, The Dynamics of Control, Birkh¨auser Boston, 2000. [30] F. Colonius and W. Kliemann, Morse decomposition and spectra on flag bundles, J. Dynamics and Differential Equations 14 (2002), 719-741. [31] F. Colonius and W. Kliemann, Dynamical systems and linear algebra, in: Hand- book of Linear Algebra (L. Hogben, ed.), CRC Press, Boca Raton, 2nd ed., 2014, 79-1 – 79-23. [32] N.D. Cong, Topological Dynamics of Random Dynamical Systems, Oxford Mathe- matical Monographs, Clarendon Press, 1997. [33] N.D. Cong, Structural stability and topological classification of continuous time, linear hyperbolic cocycles, Random Comp. Dynamics 5 (1997), 19-63. [34] C. Conley, Isolated Invariant Sets and the Morse Index, Reg. Conf. Ser. Math., vol. 38, Amer. Math. Soc., Providence, RI (1978). [35] C. Conley, The structure of a flow I, Ergodic Theory Dyn. Sys. 8 (1988), 11-26. [36] W.A. Coppel, Dichotomies in , Springer-Verlag, 1978. [37] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 1095-1113. [38] B.D. Craven, Lebesgue Measure and , Pitman, 1982. [39] J. Daleckii and M. Krein, Stability of Solutions of Differential Equations in Ba- nach Space, Amer. Math. Soc., 1974. Bibliography 273

[40] N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory, Wiley-Interscience, 1977. [41] R. Durrett, Probability: Theory and Examples, 4th ed., Cambridge University Press, Cambridge, 2010. [42] R. Easton, Geometric Methods for Discrete Dynamical Systems, Oxford University Press, 1998. [43] S. Elaydi, An Introduction to Difference Equations. Third edition, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2005. [44] D. Elliott, Bilinear Control Systems: Matrices in , Springer-Verlag, 2009. [45] R. Fabbri, R. Johnson, and L. Zampogni, Nonautonomous differential systems in two dimensions, Chapter 2 in: Handbook of Differential Equations, Ordinary Dif- ferential Equations, volume 4, F. Batelli and M. Feˇckan, eds., Elsevier 2008. [46] H. Federer, Gometric Measure Theory, Springer, New York, 1969, Reprint 2013. [47] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math., 21 (1971), 193-226. [48] R. Feres, Dynamical Systems and Semisimple Groups. An Introduction, Cambridge University Press, 1998. [49] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. 31 (1960), 457-469. [50] G. Floquet, Sur les ´equations diff´erentielles lin´eaires `acoefficientsp´eriodiques, Ann. Ecole´ Norm. Sup. 12 (1883), 47-88. [51] P. Ganssler¨ and W. Stute, Wahrscheinlichkeitstheorie, Springer-Verlag, 1977. [52] I. Gohberg, S. Goldberg, and N. Krupnik, Traces and of Linear Operators, Birkh¨auser, 2000. [53] I.Y. Goldsheid and G.A. Margulis, Lyapunov indices of a product of random matrices. Russian Mathematical Surveys 44:5(1989), 11-71. [54] M. Golubitsky and M. Dellnitz, Linear Algebra and Differential Equations, Brooks Cole Pub. Co., 1999. [55] W.H. Greub, Multilinear Algebra, Springer-Verlag, Berlin 1967, 2nd ed., 1978. [56] D.M. Grobman, On homeomorphisms of systems of differential equations,Doklady AN SSSR 128 (1959), 880-881. [57] A. Grorud and D. Talay, Approximation of Lyapunov exponents of nonlinear sto- chastic differential equations, SIAM Journal on , 56(2), (1996), 627-650. [58] L. Grune,¨ Numerical stabilization of bilinear control systems, SIAM Journal on Control and Optimization 34 (1996), 2024-2050. [59] L. Grune,¨ A uniform exponential spectrum for linear flows on vector bundles, J. Dyn. Diff. Equations, 12 (2000), 435-448. [60] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, 1983. [61] W. Hahn, Stability of Motion, Springer-Verlag, 1967. [62] P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc. 11 (1960), 610-620. [63] D.A. Harville, Matrix Algebra from a Statistician’s Perspective, Springer-Verlag, New York, 1997. 274 Bibliography

[64] E. Heil, Differentialformen und Anwendungen auf Vektoranalysis, Differentialglei- chungen, Geometrie, Bibliographisches Institut, 1974. [65] U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Springer- Verlag, 1994. [66] S. Hilger, Lineare Systeme Periodischer Differentialgleichungen, Diplomarbeit, Universit¨at W¨urzburg, 1986. [67] G.W. Hill, On the part of the motion of the lunar perigee, which is a function of the mean motions of the sun and the moon, Acta Math. 8 (1886), 1-36. [68] D. Hinrichsen and A.J. Pritchard, Mathematical Systems Theory, Springer- Verlag, 2005. [69] M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Lin- ear Algebra, Academic Press, New York, 1974. [70] M.W. Hirsch, S. Smale and R.L. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos, Elsevier, 2004. [71] L. Hogben (ed.), Handbook of Linear Algebra, 2nd ed., CRC Press, Boca Raton, 2014. [72] R.A. Horn and C.R. Johnson, Matrix Analysis, 2nd. ed., Cambridge University Press, Cambridge, 2013. [73] K. Josic´ and R. Rosenbaum, Unstable solutions of nonautonomous linear differ- ential equations, SIAM Review 50 (2008), 570-584. [74] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1984. [75] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995. [76] C. Kawan and T. Stender, Lipschitz conjugacy of linear flows, J. London Math. Soc. 80 (2009), 699-715. [77] C. Kawan and T. Stender, Growth rates for semiflows on Hausdorff spaces,J. Dynamics and Differential Equations 24 (2012), 369-390. [78] R.Z. Khasminskii, Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems, Theory of Probability and its Applications 11 (1967), 144-147. [79] R.Z. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and No- ordhoff, Alphen 1980. (Translation of the Russian edition, Nauka, Moscow 1969). [80] J.F.C. Kingman, Subadditive ergodic theory, Ann. Prob. 1 (1973), 883-904. [81] P. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathemat- ical Surveys and Monographs 176, American Mathematical Society, 2011. [82] P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equa- tions, Springer-Verlag, 2000. [83] H.-J. Kowalsky, Lineare Algebra, 4. Auflage, DeGruyter, Berlin 1969. [84] H.-J. Kowalsky and G.O. Michler, Lineare Algebra, 12. Auflage, DeGruyter, Berlin 2003. [85] U. Krause and T. Nesemann, Differenzengleichungen und Diskrete Dynamische Systeme, 2. Auflage, De Gruyter, Berlin/Boston, 2011. [86] U. Krengel, Ergodic Theorems, De Gruyter, 1985. [87] N. H. Kuiper, The topology of the solutions of a linear differential equation on Rn, Manifolds, Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973), pp. 195-203. Univ. Tokyo Press, Tokyo, 1975. Bibliography 275

[88] N.H. Kuiper, Topological conjugacy of real projective transformations, Topology 15 (1976), 13-22. [89] N.H. Kuiper, J. W. Robbin, Topological classification of linear endomorphisms, Invent. Math. 19 (1973), 83-106. [90] M. Kulczycki, Hadamard’s inequality in inner product spaces, Universitatis Iagel- lonicae Acta Mathematica, Fasc. XL (2002), 113-116. [91] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Uni- versity Press, 1990. [92] N. N. Ladis, The topological equivalence of linear flows, Differ. Equations 9 (1975), 938-947. [93] E. Lamprecht, Lineare Algebra 2, Birkh¨auser, 1983. [94] G.A. Leonov, Strange Attractors and Classical Stability Theory, St. Petersburg University Press, St. Petersburg, Russia, 2008. [95] Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dy- namical systems in a Banach space, Memoirs of the AMS, 2010. [96] F. Lowenthal, Linear Algebra with Linear Differential Equations, John Wiley & Sons, 1975. [97] A.M. Lyapunov, The General Problem of the Stability of Motion, Comm. Soc. Math. Kharkov (in Russian), 1892. Probl`eme G´eneral de la Stabilit´edeMouvement, Ann. Fac. Sci. Univ. Toulouse 9 (1907), 203-474, reprinted in Ann. Math. Studies 17, Princeton (1949), in English Taylor & Francis 1992. [98] W. Magnus and S. Winkler, Hill’s Equation, Dover Publications 1979. Corrected reprint of the 1966 edition. [99] M. Marcus, Finite Dimensional Multilinear Algebra, Part I and II, Marcel Dekker, New York 1973 and 1975. [100] L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. J. 12 (1960), 305-317. [101] W.S. Massey, A Basic Course in , Springer-Verlag, 1991. [102] R. Mennicken, On the convergence of infinite Hill-type determinants,Archivefor Rational and Analysis, 30 (1968), 12-37. [103] R. Merris, Multilinear Algebra, Gordon Breach, Amsterdam, 1997. [104] K.R. Meyer and G.R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Springer-Verlag, 1992. [105] P.D. McSwiggen, K. Meyer, Conjugate phase portraits of linear systems,Amer- ican Mathematical Monthly, 115 (2008), 596-614. [106] R.K. Miller, Almost periodic differential equations as dynamical systems with ap- plications to the existence of a.p. solutions, J. Diff. Equations, 1 (1965), 337-345. [107] K. Mischaikow, Six lectures on Conley index theory. In: Dynamical Systems, (CIME Summer School 1994), R. Johnson editor, Lecture Notes in Math., vol. 1609, Springer-Verlag, Berlin, 1995, pp. 119-207. [108] E. Oja, A simplified neuron model as a principal component analyzer, J. Math. Biol. 15(3) (1982), 267-273. [109] V.I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197-231. [110] M. Patrao˜ and L.A.B. San Martin, Semiflows on topological spaces: chain tran- sitivity and semigroups, J. Dyn. Diff. Equations 19 (2007), 155-180. 276 Bibliography

[111] M. Patrao˜ and L.A.B. San Martin, Morse decompositions of semiflows on fiber bundles, Discrete and Continuous Dynamical Systems 17 (2007), 113-139. [112] L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, 1991. [113] H. Poincare´, Sur les d´eterminants d’ordre infini, Bulletin de la Soci´et´eMathema- tiques de France, 14 (1886), 77-90. [114] M. Rasmussen, Morse decompositions of nonautonomous dynamical systems,Trans. Amer. Math. Soc., 359 (2007), 5091-5115. [115] M. Rasmussen, All-time Morse decompositions of linear nonautonomous dynamical systems, Proc. Amer. Math. Soc., 136 (2008), 1045-1055. [116] M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Sys- tems, Lecture Notes in Mathematics,vol. 1907, Springer-Verlag, 2007. [117] C. Robinson, Dynamical Systems, 2nd edition, CRC Press, 1998. [118] K.P. Rybakowski, The Index and Partial Differential Equations, Springer-Verlag, 1987. [119] R.J. Sacker and G.R. Sell, Existence of dichotomies and invariant splittings for linear differential systems I, J. Diff. Equations, 13 (1974), 429-458. [120] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc., 291 (1985), 1-41. [121] D. Salamon, E. Zehnder, Flows on vector bundles and hyperbolic sets, Trans. Amer. Math. Soc., 306 (1988), 623-649. [122] L.SanMartinandL.Seco,Morse and Lyapunov spectra and dynamics on flag bundles, Ergodic Theory and Dynamical Systems 30 (2010), 893-922. [123] B. Schmalfuß, Backward Cocycles and Attractors of Stochastic Differential Equa- tions. In International Seminar on Applied Mathematics – Nonlinear Dynamics: Approximation and Global Behaviour (1992), V. Reitmann, T. Riedrich, and N. Koksch, eds., Technische Universit¨at Dresden, pp. 185-192. [124] J. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc. 203 (1975), 259-390. [125] G. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, 2002. [126] M.A. Shayman, of the matrix Riccati equation, SIAM J. Control Optim. 24 (1986), 1-65. [127] R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King, Stability criteria for switched and hybrid systems, SIAM Review 49 (2007), 545-592. [128] C.E. Silva, Invitation to Ergodic Theory, Student Mathematical Library, Vol. 42, Amer. Math. Soc., 2008. [129] E.D. Sontag, Mathematical , Springer-Verlag, 1998. [130] J.M. Steele, Kingman’s subadditive ergodic theorem, Ann. Inst. Poincar´e, Prob. Stat. 26 (1989), 93-98. [131] J.J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems,JohnWi- ley & Sons, 1950 (reprint Wiley Classics Library 1992). [132] D. Talay, Approximation of upper Lyapunov exponents of bilinear stochastic differ- ential systems, SIAM Journal on , 28(4), (1991), 1141-1164. [133] G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Math. Vol. 149, Amer. Math. Soc., 2012. [134] H. Verdejo, L. Vargas and W. Kliemann, Stability of linear stochastic systems via Lyapunov exponents and applications to power systems, Appl. Math. Comput. 218 (2012), 11021-11032. Bibliography 277

[135] R.E. Vinograd, On a criterion of instability in the sense of Lyapunov of the so- lutions of a of ordinary differential equations, Doklady Akad. Nauk SSSR (N.S.) 84 (1952), 201-204. [136] A. Wach, A note on Hadamard’s inequality, Universitatis Iagellonicae Acta Math- ematica, Fasc. XXXI (1994), 87-92. [137] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982. [138] M.J. Ward, Industrial Mathematics, Lecture Notes, Dept. of Mathematics, Univ. of British Columbia, Vancouver, B.C., Canada, 2008 (unpublished). [139] S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Springer-Verlag, 1994. [140] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Applica- tions, Springer-Verlag, 1996. [141] F. Wirth, Dynamics of time-varying discrete-time linear systems: spectral theory and the projected system, SIAM J. Control Optim. 36 (1998), 447-487. [142] F. Wirth, A converse Lyapunov theorem for linear parameter-varying and linear switching systems, SIAM J. Control Optim. 44 (2006), 210-239. [143] W.M. Wonham, Linear Mulivariable Control: a Geometric Approach, Springer- Verlag, 1979. [144] W.-C. Xie, Dynamic Stability of Structures, Cambridge University Press, Cam- bridge, 2006.

Index

adapted norm, 35, 40 linear periodic difference equation, almost periodic function, 154 135 alternating k-linear map, 83 linear periodic differential equations, alternating product, 84 143 attractor, 159, 168 random linear system, 229, 232 and Morse decomposition, 162 chain, 50, 60 neighborhood, 159 concatenation, 186 attractor-repeller pair, 160 jump time, 51, 52, 61, 66 total time, 51 base component, 114, 121 chain component, 51, 58, 60, 62, 165 base flow, 114 and Lyapunov space, 72, 76, 92, 134, base space, 114, 121 142 basic set, 60 in Grassmannian bilinear control system, 201 linear autonomous differential binary representation, 65 equation, 92 Birkhoff’s ergodic theorem, 214 in projective bundle bit shift, 65 linear periodic difference equation, Blaschke’s theorem, 55 134 Borel-Cantelli Lemma, 256 linear periodic differential equation, 142 center subbundle topological linear flow, 173, 182 linear periodic difference equation, in projective space 135 linear autonomous difference linear periodic differential equation, equation, 75 143 linear autonomous differential random linear system, 229, 232 equation, 72 robust linear system, 205 chain exponent, 172 topological linear system, 175 chain limit set, 164 center subspace chain reachability, 51 linear autonomous difference chain recurrent point, 51, 60 equation, 24 chain recurrent set, 51, 58, 60, 165 linear autonomous differential and connectedness, 58, 61 equation, 16 and limit sets, 53

279 280 Index

chain transitive set, 51, 60 equilibrium, 31 and time reversal, 53, 61 equivalence of flows, 45 for time shift, 200 ergodic flow, 227 maximal, 51, 58, 60 ergodic map, 212 characteristic number, 99 uniquely, 221 cocycle, 114, 120 evaluation map, 116 2-parameter, 102, 118, 131 exponential growth rate linear periodic difference equation, finite time, 172 131 linear autonomous difference linear topological system, 170 equation, 20 random linear, 225, 231 linear autonomous differential conjugacy equation, 12 and chain transitivity, 54, 63 linear periodic difference equation, and fixed points, 32 128, 131, 132 and limit sets, 54 linear periodic differential equation, and Morse decompositions, 156 137, 139 and periodic solutions, 32 linear topological flow, 172, 173, 192 and structural stability, 34 of a function, 121, 123 dynamical systems, 32, 39 volume, 87, 88, 90 in projective space, 79 exterior power, 238 linear, 33 exterior product, 85 linear autonomous differential equations, 33 Fatou’s lemma, 219 linear contractions, 41 Fenichel’s uniformity lemma, 175 linear difference equations, 43 fiber, 170 projective flows, 75, 77 Fibonacci numbers, 78 smooth, 32, 33, 39 fixed point, 31 Conley index, 66 asymptotically stable, 16, 23 connected component, 57 exponentially stable, 16, 24 control system, 117 stable, 16, 23 bilinear, 171, 197 unstable, 16, 24 linear, 171 flag manifold, 95, 240 cycles, 156 metric, 240 cylinder sets, 226 flag of subspaces, 240, 247 linear autonomous difference dynamical system equation, 22 continuous, 30, 38 linear autonomous differential linear skew product, 120, 121, 231 equation, 15, 71 bilinear control system, 201 linear periodic difference equation, periodic difference equation, 131 133 robust linear system, 201 linear periodic differential equation, metric, 117, 121, 224 141 random linear, 117, 121, 230 random, 258 topological linear, 170 Floquet exponent, 128, 137 Floquet multiplier, 128, 137 eigenspace, 6 Floquet space, 132, 139 complex generalized, 6 Floquet theory, 127, 174, 228 random, 258 flow real, 9 continuous, 30 real generalized, 9, 26 ergodic, 227 eigenvalue linear skew product, 114 stability, 16, 24, 109, 128, 137 structurally stable, 34 Index 281

Fourier expansion, 148, 213 Kolmogorov’s construction, 226 fundamental domain, 41 fundamental solution limit set, 47, 49, 59 linear autonomous difference and chain component, 58 equation, 18 and chain transitivity, 53, 61 linear autonomous differential and time reversal, 50 equation, 4 linear autonomous differential equation nonautonomous linear difference on Grassmannian, 87 equation, 118 linear contraction, 40 nonautonomous linear differential linear expansion, 40 equation, 101 linear oscillator Furstenberg-Kesten Theorem, 252 autonomous, 18 periodic, 144 generalized eigenspace with periodic restoring force, 145, 150 complex, 6 with random restoring force, 263, 265 real,9,26 with uncertain , 206 , 10 with uncertain restoring force, 204, , 38, 121 207 Goldsheid-Margulis metric, 240, 247 linearized differential equation, 117 gradient-like system, 66 Liouville formula, 105 Grassmannian locally integrable matrix function, 100 linear autonomous differential Lyapunov exponent equation, 87 average, 263 metric, 85, 87, 239, 240 formula on the unit sphere, 122, 203 Gronwall’s lemma, 103 linear autonomous difference equation, 20 Hadamard inequality, 85, 94 linear autonomous differential Hamiltonian system, 143, 152 equation, 12 Hausdorff distance, 56 linear nonautonomous difference Hill’s equation, 145 equation, 119 homoclinic structures, 156 linear nonautonomous differential hyperbolic matrix, 34, 43 equation, 106 hyperellipsoid, 263 linear periodic difference equation, 131, 132 integrability condition, 227, 231 linear periodic differential equation, invariance of domain theorem, 38 139, 140, 142 invariant set, 156 linear random system isolated, 156 numerical computation, 270 metric dynamical system, 227 linear skew product flow, 115 minimal, 54 linear topological flow, 172, 173, 192, Jordan curve theorem, 209 196 Jordan normal form, 129 random linear system and smooth conjugacy, 33, 39 continuous time, 228 complex, 6 discrete time, 232 real, 7 random system, 263 Jordan subspace, 73, 77 Lyapunov space jump times, 51, 52, 61, 66 and volume growth rate, 88 linear autonomous difference equation kinematic similarity transformation, 142 in projective space, 76 kinetic energy, 152 linear autonomous differential Kingman’s subadditive ergodic equation, 13, 21 theorem, 217 in projective space, 72 282 Index

linear periodic difference equation, , 152 132 damped, 145 linear periodic differential equation, inverted, 151 139 with oscillating pivot, 150 linear topological systems, 173 periodic function, 200 nonautonomous linear differential Pl¨ucker embedding, 85 equation, 111 polar coordinates, 48 random linear system polar decomposition, 78 continuous time, 228 potential energy, 152 discrete time, 232 principal component analysis, 79 Lyapunov transformation, 122, 133, 142 principal fundamental solution, 101, 118 linear periodic difference equation, Mathieu’s equation, 145, 151 130 stability diagram, 150 linear periodic differential equation, maximal ergodic theorem, 213 136 measure preserving map, 212 probability measure metric space ergodic, 212, 215, 227 complete, 47 invariant, 211 connected, 49 projection minimal invariant set, 54 orthogonal, 235, 240 matrix, 137 projective bundle, 171 Morse decomposition, 156 metric, 171 and attractor , 162 projective flow, 70, 75 finest, 157, 165, 173, 182 projective space, 69 order, 157 metric, 69, 178 Morse sets, 156 Morse spectral interval, 173, 188 quasi-periodic function, 154 boundary points, 191, 196 Morse spectrum, 172 random linear differential equation, 225, and Lyapunov exponents, 192 229 for time reversed flow, 187 repeller, 159 periodic, 187 complementary, 160 Multiplicative Ergodic Theorem neighborhood, 159 continuous time, 227 Riccati equation, 68, 81 deterministic, 242 robust linear system, 117, 197, 201 discrete time, 231 rotation, 76, 109 multiplicity Lyapunov exponent, 111 Selgrade bundle, 173 Selgrade’s theorem, 182 Newton method, 150 semi-dynamical system, 168 normal basis, 108 semiconjugacy, 70 Oja’s flow, 79 semimartingale helix, 270 one-sided time set, 30, 39 semisimple eigenvalue, 17 Ornstein–Uhlenbeck process, 265 shift, 200 Oseledets space similarity of matrices, 32, 39, 234 continuous time, 228 simple vector, 84, 85 discrete time, 232 singular value, 228, 232, 237, 253, 263 Oseledets’ Theorem singular value decomposition, 237 continuous time, 227 exterior power, 238 discrete time, 231 skew-component, 115, 170 solution parallelepiped, 83, 106 Carath´eodory, 100, 124 Index 283

existence, 30 stochastic differential equation, 265 linear autonomous difference stochastic linear differential equation, equation, 18 226, 269 linear autonomous differential subadditive ergodic theorem, 217 equation, 4 subadditive sequence, 244 periodic, 31 subbundle, 171 solution formula exponentially separated, 189, 190 in Jordan block, 11, 12, 19, 20 in projective space, 71, 73, 77 Theon of Smyrna, 78 on the sphere, 78, 203 theorem scalar differential equation, 101 Arzel´a-Ascoli, 115 solution map Banach’s fixed point, 102 continuity, 5, 18, 104, 119 Birkhoff’s Ergodic, 214 continuity with respect to Blaschke’s, 55 parameters, 104, 119 Furstenberg-Kesten, 252 spectrum invariance of domain, 38 eigenvalues of a matrix, 6 Jordan curve, 209 Lyapunov spectrum, 204 Kingman’s subadditive ergodic, 217 Lebesgue’s on dominated Lyapunov spectrum for a random convergence, 104 system, 228 maximal ergodic, 213 Morse spectrum, 172, 204 Multiplicative Ergodic, 227, 231 uniform growth spectrum, 195 deterministic, 242 stability Oseledets’, 227, 231 and eigenvalues, 16, 24, 109, 152 Poincar´e-Bendixson, 65 asymptotic, 16, 24, 135, 143, 177 Selgrade’s, 182 exponential, 16, 24, 135, 143, 177, 205 time shift, 115, 200 almost sure, 229, 232 time-n map, 38 stability diagram time-t map, 30 linear oscillator with uncertain time-one map, 120 restoring force, 205, 207 time-reversed equation, 13 Mathieu equation, 150 linear difference equation, 21 random linear oscillator, 265 time-reversed flow, 60, 187, 194 stability radius, 206 and chain transitivity, 53, 61 stable fixed point, 16, 23 and limit sets, 50 stable subbundle, 175, 177 time-varying perturbation, 117 linear periodic difference equation, topology 135 uniform convergence, 115 linear periodic differential equation, type number, 123 143 random linear system, 229, 232 uniform growth spectrum, 195 robust linear system, 205 uniquely ergodic map, 221 stable subspace unstable subbundle, 175 linear autonomous difference linear periodic difference equation, equation, 24 135 linear autonomous differential linear periodic differential equation, equation, 16 143 linear periodic difference equation, random linear system, 229, 232 135 robust linear system, 205 linear periodic differential equation, unstable subspace 143 linear autonomous difference random linear system, 229, 232 equation, 24 284 Index

linear autonomous differential equation, 16 linear periodic difference equation, 135 linear periodic differential equation, 143 random linear system, 229, 232 variation-of-constants formula, 103 vector bundle, 170 fiber, 170 subbundle, 171 volume, 83 volume growth rate, 87, 88, 90, 106, 111

Whitney sum, 173, 182 , 105 This book provides an introduction to the inter- play between linear algebra and dynamical systems MR GSRXMRYSYW XMQI ERH MR HMWGVIXI XMQI -X ½VWX reviews the autonomous case for one matrix A via induced dynamical systems in Rd and on Grassmannian manifolds. Then the main nonauto- nomous approaches are presented for which the time dependency of A(t ) is given via skew-product ¾S[W YWMRK TIVMSHMGMX] SV XSTSPSKMGEP GLEMR VIGYVVIRGI  SV IVKSHMG TVSTIVXMIW MRZEVMERXQIEWYVIW 8LIEYXLSVWHIZIPSTKIRIVEPM^EXMSRWSJ VIEPTEVXWSJ IMKIRZEPYIW and eigenspaces as a starting point for a linear algebra for classes of time-varying linear W]WXIQWREQIP]TIVMSHMGVERHSQERHTIVXYVFIH SVGSRXVSPPIH W]WXIQW 8LIFSSOTVIWIRXWJSVXLI½VWXXMQIMRSRIZSPYQIEYRM½IHETTVSEGLZME0]ETYRSZ exponents to detailed proofs of Floquet theory, of the properties of the Morse spec- trum, and of the multiplicative ergodic theorem for products of random matrices. The main tools, chain recurrence and Morse decompositions, as well as classical ergodic theory are introduced in a way that makes the entire material accessible for beginning graduate students.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-158

GSM/158 AMS on the Web www.ams.orgwww.ams.org