Extremes and recurrence in dynamical systems

Physical dimension

Summary / review / table of contents

1 Introduction
1 1.1 A Transdisciplinary Research Area
1 1.2 Some Mathematical Ideas
4 1.3 Some Difficulties and Challenges in Studying Extremes
6 1.3.1 Finiteness of Data
6 1.3.2 Correlation and Clustering
8 1.3.3 Time Modulations and Noise
9 1.4 Extremes, Observables, and Dynamics
10 1.5 This Book
12 Acknowledgments
14 2 A Framework for Rare Events in Stochastic Processes and Dynamical Systems
17 2.1 Introducing Rare Events
17 2.2 Extremal Order Statistics
19 2.3 Extremes and Dynamics
20 3 Classical Extreme Value Theory
23 3.1 The i.i.d. Setting and the Classical Results
24 3.1.1 Block Maxima and the Generalized Extreme Value Distribution
24 3.1.2 Examples
26 3.1.3 Peaks Over Threshold and the Generalized Pareto Distribution
28 3.2 Stationary Sequences and Dependence Conditions
29 3.2.1 The Blocking Argument
30 3.2.2 The Appearance of Clusters of Exceedances
31 3.3 Convergence of Point Processes of Rare Events
32 3.3.1 Definitions and Notation
33 3.3.2 Absence of Clusters
35 3.3.3 Presence of Clusters
35 3.4 Elements of Declustering
37 4 Emergence of Extreme Value Laws for Dynamical Systems
39 4.1 Extremes for General Stationary Processes an Upgrade Motivated by Dynamics
40 4.1.1 Notation
41 4.1.2 The New Conditions
42 4.1.3 The Existence of EVL for General Stationary Stochastic Processes under Weaker Hypotheses 44 4.1.4 Proofs of Theorem 4.1.4 and Corollary
4.1.5 46 4.2 Extreme Values for Dynamically Defined Stochastic Processes
51 4.2.1 Observables and Corresponding Extreme Value Laws
53 4.2.2 Extreme Value Laws for Uniformly Expanding Systems
57 4.2.3 Example Revisited
59 4.2.4 Proof of the Dichotomy for Uniformly Expanding Maps
61 4.3 Point Processes of Rare Events
62 4.3.1 Absence of Clustering
62 4.3.2 Presence of Clustering
63 4.3.3 Dichotomy for Uniformly Expanding Systems for Point Processes
65 4.4 Conditions q(un), D3(un), Dp(un) and Decay of Correlations
66 4.5 Specific Dynamical Systems Where the Dichotomy Applies
70 4.5.1 Rychlik Systems
70 4.5.2 Piecewise Expanding Maps in Higher Dimensions
71 4.6 Extreme Value Laws for Physical Observables
72 5 Hitting and Return Time Statistics
75 5.1 Introduction to Hitting and Return Time Statistics
75 5.1.1 Definition of Hitting and Return Time Statistics
76 5.2 HTS Versus RTS and Possible Limit Laws
77 5.3 The Link Between Hitting Times and Extreme Values
78 5.4 Uniformly Hyperbolic Systems
84 5.4.1 Gibbs Measures
85 5.4.2 First HTS Theorem
86 5.4.3 Markov Partitions
86 5.4.4 Two-Sided Shifts
88 5.4.5 Hyperbolic Diffeomorphisms
89 5.4.6 Additional Uniformly Hyperbolic Examples
90 5.5 Nonuniformly Hyperbolic Systems
91 5.5.1 Induced System
91 5.5.2 Intermittent Maps
92 5.5.3 Interval Maps with Critical Points
93 5.5.4 Higher Dimensional Examples of Nonuniform Hyperbolic Systems
94 5.6 Nonexponential Laws
95 6 Extreme Value Theory for Selected Dynamical Systems
97 6.1 Rare Events and Dynamical Systems
97 6.2 Introduction and Background on Extremes in Dynamical Systems
98 6.3 The Blocking Argument for Nonuniformly Expanding Systems
99 6.3.1 Assumptions on the Invariant Measure
99 6.3.2 Dynamical Assumptions on (f , , )
99 6.3.3 Assumption on the Observable Type
100 6.3.4 Statement or Results
101 6.3.5 The Blocking Argument in One Dimension
102 6.3.6 Quantification of the Error Rates
102 6.3.7 Proof of Theorem
6.3.1 107 6.4 Nonuniformly Expanding Dynamical Systems
108 6.4.1 Uniformly Expanding Maps
108 6.4.2 Nonuniformly Expanding Quadratic Maps
109 6.4.3 One-Dimensional Lorenz Maps
110 6.4.4 Nonuniformly Expanding Intermittency Maps
110 6.5 Nonuniformly Hyperbolic Systems
113 6.5.1 Proof of Theorem
6.5.1 115 6.6 Hyperbolic Dynamical Systems
116 6.6.1 Arnold Cat Map
116 6.6.2 Lozi-Like Maps
118 6.6.3 Sinai Dispersing Billiards
119 6.6.4 Henon Maps
119 6.7 Skew-Product Extensions of Dynamical Systems
120 6.8 On the Rate of Convergence to an Extreme Value Distribution
121 6.8.1 Error Rates for Specific Dynamical Systems
123 6.9 Extreme Value Theory for Deterministic Flows
126 6.9.1 Lifting to Xh
129 6.9.2 The Normalization Constants
129 6.9.3 The Lap Number
130 6.9.4 Proof of Theorem
6.9.1 131 6.10 Physical Observables and Extreme Value Theory
133 6.10.1 Arnold Cat Map
133 6.10.2 Uniformly Hyperbolic Attractors: The Solenoid Map
137 6.11 Nonuniformly Hyperbolic Examples: the HENON and LOZI Maps
140 6.12 Extreme Value Statistics for the Lorenz 63 Model
141 7 Extreme Value Theory for Randomly Perturbed Dynamical Systems
145 7.1 Introduction
145 7.2 Random Transformations via the Probabilistic Approach: Additive Noise
146 7.2.1 Main Results
149 7.3 Random Transformations via the Spectral Approach
155 7.4 Random Transformations via the Probabilistic Approach: Randomly Applied Stochastic Perturbations
159 7.5 Observational Noise
163 7.6 Nonstationarity the Sequential Case
165 8 A Statistical Mechanical Point of View
167 8.1 Choosing a Mathematical Framework
167 8.2 Generalized Pareto Distributions for Observables of Dynamical Systems
168 8.2.1 Distance Observables
169 8.2.2 Physical Observables
172 8.2.3 Derivation of the Generalized Pareto Distribution Parameters for the Extremes of a Physical Observable
174 8.2.4 Comments
176 8.2.5 Partial Dimensions along the Stable and Unstable Directions of the Flow
177 8.2.6 Expressing the Shape Parameter in Terms of the GPD Moments and of the Invariant Measure of the System
178 8.3 Impacts of Perturbations: Response Theory for Extremes
180 8.3.1 Sensitivity of the Shape Parameter as Determined by the Changes in the Moments
182 8.3.2 Sensitivity of the Shape Parameter as Determined by the Modification of the Geometry
185 8.4 Remarks on the Geometry and the Symmetries of the Problem
188 9 Extremes as Dynamical and Geometrical Indicators
189 9.1 The Block Maxima Approach
190 9.1.1 Extreme Value Laws and the Geometry of the Attractor
191 9.1.2 Computation of the Normalizing Sequences
192 9.1.3 Inference Procedures for the Block Maxima Approach
194 9.2 The Peaks Over Threshold Approach
196 9.2.1 Inference Procedures for the Peaks Over Threshold Approach
196 9.3 Numerical Experiments: Maps Having Lebesgue Invariant Measure
197 9.3.1 Maximum Likelihood versus L-Moment Estimators
203 9.3.2 Block Maxima versus Peaks Over Threshold Methods
204 9.4 Chaotic Maps With Singular Invariant Measures
204 9.4.1 Normalizing Sequences
205 9.4.2 Numerical Experiments
208 9.5 Analysis of the Distance and Physical Observables for the HNON Map
212 9.5.1 Remarks
218 9.6 Extremes as Dynamical Indicators
218 9.6.1 The Standard Map: Peaks Over Threshold Analysis
219 9.6.2 The Standard Map: Block Maxima Analysis
220 9.7 Extreme Value Laws for Stochastically Perturbed Systems
223 9.7.1 Additive Noise
225 9.7.2 Observational Noise
229 10 Extremes as Physical Probes
233 10.1 Surface Temperature Extremes
233 10.1.1 Normal, Rare and Extreme Recurrences
235 10.1.2 Analysis of the Temperature Records
235 10.2 Dynamical Properties of Physical Observables: Extremes at Tipping Points
238 10.2.1 Extremes of Energy for the Plane Couette Flow
239 10.2.2 Extremes for a Toy Model of Turbulence
245 10.3 Concluding Remarks
247 11 Conclusions
249 11.1 Main Concepts of This Book
249 11.2 Extremes, Coarse Graining, and Parametrizations
253 11.3 Extremes of Nonautonomous Dynamical Systems
255 11.3.1 A Note on Randomly Perturbed Dynamical Systems
258 11.4 Quasi-Disconnected Attractors
260 11.5 Clusters and Recurrence of Extremes
261 11.6 Toward Spatial Extremes: Coupled Map Lattice Models
262 Appendix A Codes
265 A.1 Extremal Index
266 A.2 Recurrences Extreme Value Analysis
267 A.3 Sample Program
271 References
273 Index
293

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