Extremes and Recurrence in Dynamical Systems

Presents recent advances on the theory of extreme values that result from investigations of dynamical systems applications Written by a team of international experts, Extremes and Recurrence in Dynamical Systems presents a unique point of view on the mathematical theory of extremes and how it relates to applications in the natural and social sciences. Featuring an interdisciplinary approach to new concepts in pure and applied mathematical research, the book skillfully combines the areas of statistical mechanics, probability theory, measure theory, dynamical systems, statistical inference, geophysics, and software application. Emphasizing statistical mechanics, the book introduces robust theoretical embedding for the application of extreme value theory and modeling through dynamical systems. Extremes and Recurrence in Dynamical Systems also features: A careful examination of how a dynamical system can be taken as a generator of stochastic processes Discussions on the applications of statistical inference in the theoretical and heuristic use of extremes Several examples of analysis of extremes in a physical and geophysical context A final summary of the main results presented with a discussion of forthcoming research guidelines  An appendix with software Matlab® and Octave programming language to help readers to develop further understanding of the presented concepts Extremes and Recurrence in Dynamical Systems is ideal for academics and practitioners in pure and applied mathematics, probability theory, statistics, chaos, theoretical and applied dynamical systems, statistical mechanics, geophysical fluid dynamics, geosciences and complexity science.  INDICE: 1 Introduction 3 . 1.1 A Transdisciplinary Research Area 3 . 1.2 Some Mathematical Ideas 6 . 1.3 Some Difficulties and Challenges in Studying Extremes 8 . 1.3.1 Finiteness of Data 8 . 1.3.2 Correlation and Clustering 10 . 1.3.3 Time Modulations and Noise 11 . 1.4 Extremes, Observables, and Dynamics 12 . 1.5 This Book 14 . 2 A Framework for Rare Events 19 . 2.1 Introducing Rare Events 19 . 2.2 Extremal Order Statistics 20 . 2.3 Extremes and Dynamics 22 . 3 Classical Extreme Value Theory 25 . 3.1 The i.i.d. Setting and the Classical Results 26 . 3.1.1 Block Maxima and the Generalized Extreme Value Distribution 26 . 3.1.2 Examples 29 . 3.1.3 Peaks Over Threshold and the Generalised Pareto Distribution 30 . 3.2 Stationary Sequences and Dependence Conditions 31 . 3.2.1 The Blocking Argument 32 . 3.2.2 The Appearance of clusters of Exceedances 33 . 3.3 Convergence of Point Processes of Rare Events 35 . 3.3.1 Definitions and Notation 36 . 3.3.2 Absence of Clusters 37 . 3.3.3 Presence of Clusters 38 . 3.4 Elements of Declustering 40 . 4 Emergence of Extreme Value Laws for Dynamical Systems 41 . 4.1 Extremes for General Stationary Processes an Upgrade Motivated by Dynamics 42 . 4.1.1 Notation 42 . 4.1.2 The New Conditions 44 . 4.1.3 The Existence of EVL for General Stationary Stochastic Processes under Weaker Hypotheses 46 . 4.1.4 Proofs of Theorem 4.1.4 and Corollary 4.1.5 48 . 4.2 Extreme Values for Dynamically Defined Stochastic Processes 53 . 4.2.1 Observables and Corresponding Extreme Value Laws 55 . 4.2.2 Extreme Value Laws for Uniformly Expanding Systems 59 . 4.2.3 Example 4.2.1 revisited 61 . 4.2.4 Proof of the Dichotomy for Uniformly Expanding Maps 63 . 4.3 Point Processes of Rare Events 64 . 4.3.1 Absence of Clustering 64 . 4.3.2 Presence of Clustering 65 . 4.3.3 Dichotomy for Uniformly Expanding Systems for Point Processes 67 . 4.4 Conditions Äq(un), D3(un), Dp(un)  and Decay of Correlations 68 . 4.5 Specific Dynamical Systems where the Dichotomy Applies 71 . 4.5.1 Rychlik Systems 72 . 4.5.2 Piecewise Expanding Maps in Higher Dimensions 73 . 4.6 Extreme Value Laws for Physical Observables 74 . 5 Hitting and Return time Statistics 77 . 5.1 Introduction to Hitting and Return Time Statistics 77 . 5.1.1 Definition of Hitting and Return Time Statistics 78 . 5.2 HTS vs RTS and Possible Limit Laws 79 . 5.3 The Link between Hitting Times and Extreme Values 80 . 5.4 Uniformly Hyperbolic Systems 86 . 5.4.1 Gibbs Measures 87 . 5.4.2 First HTS theorem 87 . 5.4.3 Markov partitions 88 . 5.4.4 Two–sided Shifts 90 . 5.4.5 Hyperbolic Diffeomorphisms 90 . 5.4.6 Additional Uniformly Hyperbolic Examples 92 . 5.5 Non–uniformly Hyperbolic Systems 92 . 5.5.1 Induced System 93 . 5.5.2 Intermittent Maps 94 . 5.5.3 Interval Maps with Critical Points 95 . 5.5.4 Higher Dimensional Examples of Non–uniform Hyperbolic Systems 96 . 5.6 Non–exponential Laws 97 . 6 Extreme Value Theory for Selected Dynamical Systems 99 . 6.1 Rare Events and Dynamical Systems 99 . 6.2 Introduction and Background on Extremes in Dynamical Systems 100 . 6.3 The Blocking Argument for Non–uniformly Expanding Systems 101 . 6.3.1 The Blocking Argument in One Dimension 103 . 6.3.2 Quantification of the Error Rates 104 . 6.3.3 Proof of Theorem 6.3.1 109 . 6.4 Non–uniformly Expanding Dynamical Systems 110 . 6.4.1 Uniformly Expanding Maps 110 . 6.4.2 Non–uniformly Expanding Quadratic Maps 111 . 6.4.3 One–dimensional Lorenz Maps 111 . 6.4.4 Non–uniformly Expanding Intermittency Maps. 112 . 6.5 Non–uniformly Hyperbolic Systems 114 . 6.5.1 Proof of Theorem 6.5.1 117 . 6.6 Hyperbolic Dynamical Systems 118 . 6.6.1 Arnold Cat Map 118 . 6.6.2 Lozi–like Maps 119 . 6.6.3 Sinai Dispersing Billiards 121 . 6.6.4 Hénon maps 121 . 6.7 Skew–product Extensions of Dynamical Systems. 122 . 6.8 On the Rate of Convergence to an Extreme Value Distribution 123 . 6.8.1 Error Rates for Specific Dynamical Systems 126 . 6.9 Extreme Value Theory for Deterministic Flows 128 . 6.9.1 Lifting to Xh 131 . 6.9.2 The Normalization Constants 132 . 6.9.3 The Lap Number 132 . 6.9.4 Proof of Theorem 6.9.1. 133 . 6.10 Physical Observables and Extreme Value Theory 135 . 6.10.1 Arnold Cat Map 135 . 6.10.2 Uniformly HyperbolicHyperbolic Attractors: the Solenoid Map 139 . 6.11 Non–uniformly Hyperbolic Examples: the Hénon and Lozi maps 142 . 6.11.1 Extreme Value Statistics for the Lorenz 63 Model 143 . 7 Extreme Value Theory for Randomly Perturbed Dynamical Systems 147 . 7.1 Introduction 147 . 7.2 Random Transformations via the Probabilistic Approach: Additive Noise 148 . 7.2.1 Main Results 151 . 7.3 Random Transformations via the Spectral Approach 157 . 7.4 Random Transformations via the Probabilistic Approach: Randomly Applied Stochastic Perturbations 160 . 7.5 Observational Noise 164 . 7.6 Non–stationarity the Sequential Case 167 . 8 A Statistical Mechanical Point of View 169 . 8.1 Choosing a Mathematical Framework 169 . 8.2 Generalized Pareto Distributions for Observables of Dynamical Systems 170 . 8.2.1 Distance Observables 171 . 8.2.2 Physical Observables 174 . 8.2.3 Derivation of the Generalised Pareto Distribution Parameters for the Extremes of a Physical Observable 176 . 8.2.4 Comments 178 . 8.2.5 Partial Dimensions along the Stable and Unstable Directions of the Flow 179 . 8.2.6 Expressing the shape parameter in terms of the GPD moments and of the invariant measure of the system 180 . 8.3 Impacts of Perturbations: Response Theory for Extremes 182 . 8.3.1 Sensitivity of the Shape Parameter as Determined by the Changes in the Moments 184 . 8.3.2 Sensitivity of the shape parameter as determined by the modification of the geometry 187 . 8.4 Remarks on the Geometry and the Symmetries of the Problem 190 . 9 Extremes as Dynamical and Geometrical Indicators 193 . 9.1 The Block Maxima Approach 194 . 9.1.1 Extreme Value Laws and the Geometry of the Attractor 195 . 9.1.2 Computation of the Normalizing Sequences 196 . 9.1.3 Inference Procedures for the Block Maxima Approach 198 . 9.2 The Peaks Over Threshold Approach 200 . 9.3 Numerical Experiments: Maps Having Lebesgue Invariant Measure 202 . 9.3.1 Maximum Likelihood vs L–moment Estimators 207 . 9.3.2 Block Maxima vs Peaks Over Threshold Methods 208 . 9.4 Chaotic Maps with Singular Invariant Measures 209 . 9.4.1 Nomalizing Sequences 209 . 9.4.2 Numerical Experiments 211 . 9.5 Analysis of the Distance and Physical Observables for the Hénon map 218 . 9.5.1 Remarks 221 . 9.6 Extremes as Dynamical Indicators 222 . 9.6.1 The Standard Map: Peaks Over Threshold Analysis 223 . 9.6.2 The Standard Map: Block Maxima Analysis 224 . 9.7 Extreme Value Laws for Stochastically Perturbed Systems 228 . 9.7.1 Additive Noise 229 . 9.7.2 Observational Noise 232 . 10 Extremes as Physical Probes 235 . 10.1 Surface Temperature Extremes 235 . 10.1.1 Normal, Rare and Extreme Recurrences 236 . 10.1.2 Analysis of the Temperature Records 237 . 10.2 Dynamical Properties of Physical Observables: Extremes at Tipping Points 240 . 10.2.1 Extremes of Energy for the Plane Couette Flow 241 . 10.2.2 Extremes for a Toy Model of Turbulence 247 . 10.3 Concluding Remarks 249 . 11 Conclusions 251 . 11.1 Main Concepts of This Book 251 . 11.2 Extremes, Coarse Graining, and Parametrizations 255 . 11.3 Extremes of Non–Autonomous Dynamical Systems 257 . 11.3.1 A note on Randomly Perturbed Dynamical Systems 261 . 11.4 Quasi–disconnected Attractors 263 . 11.5 Clusters and Recurrence of Extremes 264 . 11.6 Towards Spatial Extremes: Coupled Map Lattice Models 265 . A Codes 267 . A.1 Extremal index 267 . A.2 Recurrences – Extreme Value Analysis 269 . A.3 Sample Program 273 . Index 275

• ISBN: 978-1-118-63219-2
• Editorial: Wiley–Blackwell
• Encuadernacion: Cartoné
• Páginas: 304
• Fecha Publicación: 22/04/2016
• Nº Volúmenes: 1
• Idioma: Inglés