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American Mathematical Society

Fritz Colonius Wolfgang Kliemann

Graduate Studies in Mathematics

Volume 158

Dynamical Systems and Linear Algebra


Fritz ColoniusWolfgang Kliemann

American Mathematical SocietyProvidence, Rhode Island

Graduate Studies in Mathematics

Volume 158

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

EDITORIAL COMMITTEE

Dan AbramovichDaniel 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, visitwww.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. Algebras, Linear. 2. Topological dynamics. I. Kliemann, Wolfgang. II. Title.

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This book is dedicated to the Institut fur Dynamische Systeme at UniversitatBremen, which had a lasting influence on our mathematical thinking, as wellas to our students. This book would not have been possible without theinteraction 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 Equations 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


Chapter 3. Chain Transitivity for Dynamical Systems 47

§3.1. Limit 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


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


Chapter 5. Linear Systems on Grassmannians 81

§5.1. Some Notions and Results from Multilinear Algebra 82

§5.2. Linear Systems on Grassmannians and Volume Growth 86

§5.3. Exercises 94


Part 2. Time-Varying Matrices and Linear Skew ProductSystems

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


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 Equation 144



Chapter 8. Morse Decompositions of Dynamical Systems 155

§8.1. Morse Decompositions 155

§8.2. Attractors 159

§8.3. Morse Decompositions, Attractors, and Chain Transitivity 164


Contents ix


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 ControlSystems 197



Chapter 10. Tools from Ergodic Theory 211

§10.1. Invariant Measures 211

§10.2. Birkhoff’s Ergodic Theorem 214

§10.3. Kingman’s Subadditive Ergodic Theorem 217



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-MargulisMetric 237

§11.4. The Deterministic Multiplicative Ergodic Theorem 242

§11.5. The Furstenberg-Kesten Theorem and Proof of the MET inDiscrete Time 252

§11.6. The Random Linear Oscillator 263



Bibliography 271

Index 279

Introduction

Background

Linear algebra plays a key role in the theory of dynamical systems, andconcepts 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 matrix A gives rise to a discrete timedynamical system xk+1 = Axk, k = 0, 1, 2, . . . and to a continuous timedynamical system via the linear ordinary differential equation x = Ax.

The (real) Jordan form of the matrix A allows us to write the solutionof the differential equation x = Ax explicitly in terms of the matrix ex-ponential, and hence the properties of the solutions are intimately relatedto the properties of the matrix A. Vice versa, one can consider propertiesof a linear flow in Rd and infer characteristics of the underlying matrix A.Going one step further, matrices also define (nonlinear) systems on smoothmanifolds, such as the sphere Sd−1 in Rd, the Grassmannian manifolds, theflag manifolds, or on classical (matrix) Lie groups. Again, the behavior ofsuch 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 of x = A(t)x it is not sufficientthat for all t ∈ R the matrices A(t) have only eigenvalues with negativereal part (see, e.g., Hahn [61], Chapter 62). Classical Floquet theory (seeFloquet’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 generaltime 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 basicissues for measurable linear systems with stationary time dependencies, andthe Morse spectrum together with Selgrade’s theorem [124] goes a long wayin describing the situation for continuous linear systems with chain transitivetime dependencies.

A third important area of interplay between dynamics and linear algebraarises in the linearization of nonlinear systems about fixed points or arbitrarytrajectories. Linearization of a differential equation y = f(y) in Rd about afixed point y0 ∈ Rd results in the linear differential equation x = f ′(y0)x andtheorems of the type Grobman-Hartman (see, e.g., Bronstein and Kopan-skii [21]) resolve the behavior of the flow of the nonlinear equation locallyaround y0 up to conjugacy, with similar results for dynamical systems overa stochastic or chain recurrent base.

These observations have important applications in the natural sciencesand in engineering design and analysis of systems. Specifically, they arethe basis for stochastic bifurcation theory (see, e.g., Arnold [6]), and robuststability and stabilizability (see, e.g., Colonius and Kliemann [29]). Stabil-ity radii (see, e.g., Hinrichsen and Pritchard [68]) describe the amount ofperturbation the operating point of a system can sustain while remainingstable, and stochastic stability characterizes the limits of acceptable noisein a system, e.g., an electric power system with a substantial component ofwind 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. Thereare a number of other books emphasizing these relations. In particular, wewould like to mention the book [69] by M.W. Hirsch and S. Smale, whichalways has been a great source of inspiration for us. However, this bookrestricts attention to autonomous equations. The same is true for otherbooks like M. Golubitsky and M. Dellnitz [54] or F. Lowenthal [96], whichis designed to serve as a text for a first course in linear algebra, and therelations to linear autonomous differential equations are established on anelementary level only.

Our goal is to review the autonomous case for one d × d matrix A viainduced dynamical systems in Rd and on Grassmannians, and to presentthe 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 arepresented throughout the text.

Originally, we had also planned to include some basic concepts for thestudy of genuinely nonlinear systems via linearization, emphasizing invari-ant manifolds and Grobman-Hartman type results that compare nonlinearbehavior locally to the behavior of associated linear systems. We decided toskip this discussion, since it would increase the length of this book consider-ably and, more importantly, there are excellent treatises of these problemsavailable in the literature, e.g., Robinson [117] for linearization at fixedpoints, 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 andsemigroups where many concepts have natural counterparts. The mono-graph [48] by R. Feres provides an excellent introduction. We also do nottreat nonautonomous differential equations via pullback or other fiberwiseconstructions; see, e.g., Crauel and Flandoli [37], Schmalfuß [123], and Ras-mussen [116]; our emphasis is on the treatment of families of nonautonomousequations. Further references are given at the end of the chapters.

Finally, it should be mentioned that all concepts and results in this bookcan be formulated in continuous and in discrete time. However, sometimesresults 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 libertyto 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 inChapter 5 on induced systems on Grassmannians are formulated and derivedonly in continuous time. More importantly, the proof of the multiplicativeergodic 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 9and the results on Morse decompositions in Chapter 8, which are used forits 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 beginninggraduate students by providing complete proofs of the major results in atleast one typical situation. In particular, the results on the Morse spectrumin 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 currentliterature.

xiv Introduction

Prerequisites

The reader should have basic knowledge of real analysis (including met-ric spaces) and linear algebra. No previous exposure to ordinary differentialequations is assumed, although a first course in linear differential equationscertainly is helpful. Multilinear algebra shows up in two places: in Section5.2 we discuss how the volumes of parallelepipeds grow under the flow ofa linear autonomous differential equation, which we relate to chain recur-rent sets of the induced flows on Grassmannians. The necessary elements ofmultilinear algebra are presented in Section 5.1. In Chapter 11 the proof ofthe multiplicative ergodic theorem requires further elements of multilinearalgebra which are provided in Section 11.3. Understanding the proofs inChapter 10 on ergodic theory and Chapter 11 on random linear dynamicalsystems also requires basic knowledge of σ-algebras and probability mea-sures (actually, a detailed knowledge of Lebesgue measure should suffice).The results and methods in the rest of the book are independent of theseadditional prerequisites.

Acknowledgements

The idea for this book grew out of the preparations for Chapter 79 inthe Handbook of Linear Algebra [71]. Then WK gave a course “Dynamicsand Linear Algebra” at the Simposio 2007 of the Sociedad de Matematica deChile. FC later taught a course on the same topic at Iowa State Universitywithin the 2008 Summer Program for Graduate Students of the Instituteof Mathematics and Its Applications, Minneapolis. Parts of the manuscriptwere also used for courses at the University of Augsburg in the summersemesters 2010 and 2013 and at Iowa State University in Spring 2011. Wegratefully 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 (Section4.1), Patrick Roocks (Section 5.2); Florian Ecker and Julia Rudolph (Section7.3.); and Humberto Verdejo (Section 11.6). Thanks are also due to PhilippDuren, Julian Braun, and Justin Peters. We are particularly indebted toChristoph Kawan who has read the whole manuscript and provided us withlong lists of errors and inaccuracies. Special thanks go to Ina Mette of theAMS for her interest in this project and her continuous support during thelast few years, even when the text moved forward very slowly.

The authors welcome any comments, suggestions, or corrections you mayhave.

Fritz Colonius Wolfgang Kliemann

Institut fur Mathematik Department of Mathematics

Universitat Augsburg Iowa State University

Notation

Throughout this text we will use the following notation:

A�B = (A \B) ∪ (B \A) , the symmetric difference of setsf−1(E) = {x | f(x) ∈ E} for a map f : X → Y and E ⊂ YEc the complement of a subset E ⊂ X, Ec = X \ EIE the characteristic function of a set E, IE(x) := 1

if x ∈ E and IE(x) := 0 elsewheref+(x) = max(f(x), 0), the positive part of f : X → Rlog+ x log+ x := log x for x ≥ 1 and log+ x := 0 for x ≤ 1gl(d,R), gl(d,C) the set of real (complex) d× d matricesGl(d,R), Gl(d,C) the set of invertible real (complex) d× d matricesA� the transpose of a matrix A ∈ gl(d,R)‖·‖ a norm on Rd or an induced matrix normspec(A) the set of eigenvalues μ ∈ C of a matrix AimA, trA the image and the trace of a linear map A, resp.lim sup, lim inf limit superior, limit inferiorN, N0 the set of natural numbers excluding and including 0ı ı =

√−1

z the complex conjugate of z ∈ CA A = (aij) for A = (aij) ∈ gl(d,C)Pd−1 the real projective space Pd−1 = RPd−1

Gk(d) the kth Grassmannian of Rd

L(λ) the Lyapunov space associated with a Lyapunov exponent λE 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 xdiamE = sup{d(x, y) | x, y ∈ E}, the diameter of Edist(x,E) = inf {d(x, y) | y ∈ E} , the distance of x to Ecl E, intE the topological closure and interior of E, resp.

xv

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Index

adapted norm, 35, 40

almost periodic function, 154alternating k-linear map, 83

alternating product, 84attractor, 159, 168

and Morse decomposition, 162

neighborhood, 159attractor-repeller pair, 160

base component, 114, 121

base flow, 114base space, 114, 121

basic set, 60bilinear control system, 201

binary representation, 65Birkhoff’s ergodic theorem, 214

bit shift, 65Blaschke’s theorem, 55Borel-Cantelli Lemma, 256

center subbundle

linear periodic difference equation,135

linear periodic differential equation,143

random linear system, 229, 232robust linear system, 205

topological linear system, 175center subspace

linear autonomous differenceequation, 24

linear autonomous differentialequation, 16


linear periodic differential equations,143

random linear system, 229, 232

chain, 50, 60

concatenation, 186

jump time, 51, 52, 61, 66

total time, 51

chain component, 51, 58, 60, 62, 165

and Lyapunov space, 72, 76, 92, 134,142

in Grassmannian


in projective bundle


linear periodic differentialequation, 142

topological linear flow, 173, 182

in projective space



chain exponent, 172

chain limit set, 164

chain reachability, 51

chain recurrent point, 51, 60

chain recurrent set, 51, 58, 60, 165

and connectedness, 58, 61

and limit sets, 53

279

280 Index

chain transitive set, 51, 60and time reversal, 53, 61for time shift, 200maximal, 51, 58, 60

characteristic number, 99cocycle, 114, 120

2-parameter, 102, 118, 131linear periodic difference equation,

131linear topological system, 170random linear, 225, 231

conjugacyand chain transitivity, 54, 63and fixed points, 32and limit sets, 54and Morse decompositions, 156and periodic solutions, 32and structural stability, 34dynamical systems, 32, 39in projective space, 79linear, 33linear autonomous differential

equations, 33linear contractions, 41linear difference equations, 43projective flows, 75, 77smooth, 32, 33, 39

Conley index, 66connected component, 57control system, 117

bilinear, 171, 197linear, 171

cycles, 156cylinder sets, 226

dynamical systemcontinuous, 30, 38linear skew product, 120, 121, 231

bilinear control system, 201periodic difference equation, 131robust linear system, 201

metric, 117, 121, 224random linear, 117, 121, 230topological linear, 170

eigenspace, 6complex generalized, 6random, 258real, 9real generalized, 9, 26

eigenvaluestability, 16, 24, 109, 128, 137

equilibrium, 31equivalence of flows, 45ergodic flow, 227ergodic map, 212

uniquely, 221evaluation map, 116exponential growth rate

finite time, 172linear autonomous difference

equation, 20linear autonomous differential

equation, 12linear periodic difference equation,

128, 131, 132linear periodic differential equation,

137, 139linear topological flow, 172, 173, 192of a function, 121, 123volume, 87, 88, 90

exterior power, 238exterior product, 85

Fatou’s lemma, 219Fenichel’s uniformity lemma, 175fiber, 170Fibonacci numbers, 78fixed point, 31

asymptotically stable, 16, 23exponentially stable, 16, 24stable, 16, 23unstable, 16, 24

flag manifold, 95, 240metric, 240

flag of subspaces, 240, 247linear autonomous difference


equation, 15, 71linear periodic difference equation,

133linear periodic differential equation,

141random, 258

Floquet exponent, 128, 137Floquet multiplier, 128, 137Floquet space, 132, 139Floquet theory, 127, 174, 228flow

continuous, 30ergodic, 227linear skew product, 114structurally stable, 34

Index 281

Fourier expansion, 148, 213fundamental domain, 41fundamental solution



nonautonomous linear differenceequation, 118

nonautonomous linear differentialequation, 101

Furstenberg-Kesten Theorem, 252

generalized eigenspacecomplex, 6real, 9, 26

generalized eigenvector, 10generator, 38, 121Goldsheid-Margulis metric, 240, 247gradient-like system, 66Grassmannian


metric, 85, 87, 239, 240Gronwall’s lemma, 103

Hadamard inequality, 85, 94Hamiltonian system, 143, 152Hausdorff distance, 56Hill’s equation, 145homoclinic structures, 156hyperbolic matrix, 34, 43hyperellipsoid, 263

integrability condition, 227, 231invariance of domain theorem, 38invariant set, 156

isolated, 156metric dynamical system, 227minimal, 54

Jordan curve theorem, 209Jordan normal form, 129

and smooth conjugacy, 33, 39complex, 6real, 7

Jordan subspace, 73, 77jump times, 51, 52, 61, 66

kinematic similarity transformation, 142kinetic energy, 152Kingman’s subadditive ergodic

theorem, 217

Kolmogorov’s construction, 226

limit set, 47, 49, 59and chain component, 58and chain transitivity, 53, 61and time reversal, 50

linear autonomous differential equationon Grassmannian, 87

linear contraction, 40linear expansion, 40linear oscillator

autonomous, 18periodic, 144with periodic restoring force, 145, 150with random restoring force, 263, 265with uncertain damping, 206with uncertain restoring force, 204,

207linearized differential equation, 117Liouville formula, 105locally integrable matrix function, 100Lyapunov exponent

average, 263formula on the unit sphere, 122, 203linear autonomous difference


equation, 12linear nonautonomous difference

equation, 119linear nonautonomous differential

equation, 106linear periodic difference equation,

131, 132linear periodic differential equation,

139, 140, 142linear random system

numerical computation, 270linear skew product flow, 115linear topological flow, 172, 173, 192,

196random linear system

continuous time, 228discrete time, 232

random system, 263Lyapunov space

and volume growth rate, 88linear autonomous difference equation

in projective space, 76linear autonomous differential

equation, 13, 21in projective space, 72

282 Index



linear topological systems, 173nonautonomous linear differential

equation, 111random linear system


Lyapunov transformation, 122, 133, 142

Mathieu’s equation, 145, 151stability diagram, 150

maximal ergodic theorem, 213measure preserving map, 212metric space

complete, 47connected, 49

minimal invariant set, 54monodromy matrix, 137Morse decomposition, 156

and attractor sequence, 162finest, 157, 165, 173, 182order, 157

Morse sets, 156Morse spectral interval, 173, 188

boundary points, 191, 196Morse spectrum, 172

and Lyapunov exponents, 192for time reversed flow, 187periodic, 187

Multiplicative Ergodic Theoremcontinuous time, 227deterministic, 242discrete time, 231

multiplicityLyapunov exponent, 111

Newton method, 150normal basis, 108

Oja’s flow, 79one-sided time set, 30, 39Ornstein–Uhlenbeck process, 265Oseledets space


Oseledets’ Theoremcontinuous time, 227discrete time, 231

parallelepiped, 83, 106

pendulum, 152damped, 145inverted, 151with oscillating pivot, 150

periodic function, 200Plucker embedding, 85polar coordinates, 48polar decomposition, 78potential energy, 152principal component analysis, 79principal fundamental solution, 101, 118



probability measureergodic, 212, 215, 227invariant, 211

projectionorthogonal, 235, 240

projective bundle, 171metric, 171

projective flow, 70, 75projective space, 69

metric, 69, 178

quasi-periodic function, 154

random linear differential equation, 225,229

repeller, 159complementary, 160neighborhood, 159

Riccati equation, 68, 81robust linear system, 117, 197, 201rotation, 76, 109

Selgrade bundle, 173Selgrade’s theorem, 182semi-dynamical system, 168semiconjugacy, 70semimartingale helix, 270semisimple eigenvalue, 17shift, 200similarity of matrices, 32, 39, 234simple vector, 84, 85singular value, 228, 232, 237, 253, 263singular value decomposition, 237

exterior power, 238skew-component, 115, 170solution

Caratheodory, 100, 124

Index 283

existence, 30



periodic, 31

solution formula

in Jordan block, 11, 12, 19, 20in projective space, 71, 73, 77

on the sphere, 78, 203

scalar differential equation, 101

solution map

continuity, 5, 18, 104, 119

continuity with respect toparameters, 104, 119

spectrumeigenvalues of a matrix, 6

Lyapunov spectrum, 204

Lyapunov spectrum for a randomsystem, 228

Morse spectrum, 172, 204

uniform growth spectrum, 195

stability

and eigenvalues, 16, 24, 109, 152asymptotic, 16, 24, 135, 143, 177

exponential, 16, 24, 135, 143, 177, 205

almost sure, 229, 232

stability diagram

linear oscillator with uncertainrestoring force, 205, 207

Mathieu equation, 150

random linear oscillator, 265stability radius, 206

stable fixed point, 16, 23

stable subbundle, 175, 177




robust linear system, 205

stable subspace






stochastic differential equation, 265stochastic linear differential equation,

226, 269subadditive ergodic theorem, 217subadditive sequence, 244subbundle, 171

exponentially separated, 189, 190

Theon of Smyrna, 78theorem

Arzela-Ascoli, 115Banach’s fixed point, 102Birkhoff’s Ergodic, 214Blaschke’s, 55Furstenberg-Kesten, 252invariance of domain, 38Jordan curve, 209Kingman’s subadditive ergodic, 217Lebesgue’s on dominated

convergence, 104maximal ergodic, 213Multiplicative Ergodic, 227, 231

deterministic, 242Oseledets’, 227, 231Poincare-Bendixson, 65Selgrade’s, 182

time shift, 115, 200time-n map, 38time-t map, 30time-one map, 120time-reversed equation, 13

linear difference equation, 21time-reversed flow, 60, 187, 194

and chain transitivity, 53, 61and limit sets, 50

time-varying perturbation, 117topology

uniform convergence, 115type number, 123

uniform growth spectrum, 195uniquely ergodic map, 221unstable subbundle, 175



random linear system, 229, 232robust linear system, 205

unstable subspacelinear autonomous difference

equation, 24

284 Index





variation-of-constants formula, 103vector bundle, 170

fiber, 170subbundle, 171

volume, 83volume growth rate, 87, 88, 90, 106, 111

Whitney sum, 173, 182Wronskian, 105

GSM/158

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This book provides an introduction to the inter-play between linear algebra and dynamical systems �� 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 �� and eigenspaces as a starting point for a linear algebra for classes of time-varying linear ��

�� !�� "��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.

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