Colonius F. Dynamical systems and linear algebra / F.Colonius, W.Kliemann. - Providence: American mathematical society, 2014. - xv, 284 p.: ill. - (Graduate studies in mathematics; vol.158). - Bibliogr.: p.271-277. - Ind.: p.279-284. - ISBN 978-0-8218-8319-8  Место хранения:   02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / 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 §2.5 Orientation, Notes and References ........................ 43 Chapter 3 Chain Transitivity for Dynamical Systems ............ 47 §3.1 Limit Sets and Chain Transitivity ........................ 47 §3.2 The Chain Recurrent Set .................................. 54 §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 Multilinear Algebra ........ 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 Equation .................................... 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 Attractors .............................................. 159 §8.3 Morse Decompositions, Attractors, and Chain Transitivity ............................................ 164 §8.4 Exercises ............................................... 166 §8.5 Origination, 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 Invariant 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