Holdings: Lectures on fractal geometry and dynamical systems

$Lectures on fractal geometry and dynamical systems$

Lectures on fractal geometry and dynamical systems:

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Bibliographic Details
Main Author:	Pesin, Yakov B. 1946- (Author)
Format:	Book
Language:	English
Published:	Providence, RI American Math. Soc. [u.a.] 2009
Series:	Student mathematical library 52
Subjects:	Fractals Dynamics
Online Access:	Inhaltsverzeichnis
Item Description:	Includes bibliographical references and index
Physical Description:	XVI, 314 S. Ill., graph. Darst.
ISBN:	9780821848890

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adam_text	Contents Foreword: MASS and REU at Penn State University xi Preface xiii Chapter 1. Basic Concepts and Examples 1 Lecture 1 1 a. A threefold cord: fractals, dynamics, and chaos 1 b. Fractals: intricate geometry and self-similarity 2 с Dynamics: things that move (or don t) 7 Lecture 2 10 a. Dynamical systems: terminology and notation 10 b. Population models and the logistic map 13 Lecture 3 19 a. A linear map with chaotic behaviour and the middle-third Cantor set 19 b. The Cantor set and symbolic dynamics 24 Lecture 4 28 a. Some point-set topology 28 b. Metric spaces 31 с Lebesgue measure 34 Lecture 5 37 a. The topologica! structure of symbolic space and the Cantor set 37 b. What the coding map doesn t do 41 с Geometry of Cantor sets 42 Lecture 6 46 a. More general constructions 46 b. Making sense of it all 50 Chapter 2. Fundamentals of Dimension Theory 53 Lecture 7 53 a. Definition of Hausdorff dimension 53 b. Hausdorff dimension of the middle-third Cantor set 58 с Alternative definitions of Hausdorff dimension 60 Lecture 8 62 a. Properties of Hausdorff dimension 62 b. Topological dimension 66 Lecture 9 68 a. Comparison of Hausdorff and topological dimension 68 b. Metrics and topologies 71 с Topology and dimension 74 Lecture 10 .75 a. Hausdorff dimension of Cantor sets 75 b. Moran s Theorem 76 с Moran constructions 80 d. Dynamical constructions and iterated function systems 82 Lecture 11 85 a. Box dimension: another way of measuring dimension 85 b. Properties of box dimension 88 Lecture 12 90 a. Relationships between the various dimensions 90 b. A counterexample 94 с Stability and subadditivity 98 Chapter 3. Measures: Definitions and Examples 101 Lecture 13 101 a. A little bit of measure theory 101 b. Lebesgue measure and outer measures 105 c. Hausdorff measures 109 Lecture 14 110 a. Choosing a good outer measure 110 b. Bernoulli measures on symbolic space 111 с Measures on Cantor sets 113 d. Markov measures 114 Lecture 15 117 a. The support of a measure 117 b. Subshifts of finite type: One-dimensional Markov maps 120 Chapter 4. Measures and Dimensions 123 Lecture 16 123 a. The Uniform Mass Distribution Principle: Using measures to determine dimension 123 b. Pointwise dimension and the Non-uniform Mass Distribution Principle 125 Lecture 17 128 a. Variable pointwise dimension 128 b. Hausdorff dimension of exact dimensional measures 135 с Pointwise dimension of Hausdorff measures 137 Lecture 18 138 a. Local entropy 138 b. Kolmogorov-Sinai entropy 142 с Topologica! entropy 143 Lecture 19 146 a. Entropy of Markov measures 146 b. Hausdorff dimension of Markov constructions 149 Lecture 20 151 a. Lyapunov exponents 151 b. Fractals within fractals 155 Chapter 5. Discrete-Time Systems: The FitzHugh-Nagumo Model 159 Lecture 21 159 a. The FitzHugh-Nagumo model for neurons 159 b. Numerical investigations: From continuous to discrete 164 Lecture 22 167 a. Studying the local map 167 b. Stability of fixed points for general maps 169 Lecture 23 175 a. Stability of fixed points for the FitzHugh-Nagumo model 175 b. Periodic points 177 Lecture 24 181 a. Beyond period-doubling: Down the rabbit hole 181 b. Becoming one-dimensional 185 Chapter 6. The Bifurcation Diagram for the Logistic Map 191 Lecture 25 191 a. Bifurcations of the logistic map 191 b. Classifying bifurcations 194 Lecture 26 198 a. The period-doubling cascade 198 b. Chaos at the end of the bifurcation diagram 199 с The centre cannot hold: escape to infinity 202 Lecture 27 204 a. Finding the relevant part of phase space: ω -limit sets 204 b. Windows of stability in the bifurcation diagram 206 с Chaos outside the windows of stability 207 Chapter 7. Chaotic Attractors and Persistent Chaos 209 Lecture 28 209 a. Trapping regions 209 b. Attractors 212 Lecture 29 215 a. The Smale- Williams solenoid 215 b. Uniform hyperbolicity 218 c. Symbolic dynamics 221 Lecture 30 224 a. Dimension of direct products 224 b. Quantifying the attractor 228 c. Lyapunov exponents in multiple dimensions 229 d. The non-conformal case 231 e. The attractor for the FitzHugh-Nagumo map 232 Chapter 8. Horseshoes and Intermittent Chaos 233 Lecture 31 233 a. The Smale horseshoe: A trapping region that isn t 233 b. Hausdorff dimension of the horseshoe 237 с Symbolic dynamics on the Smale horseshoe 239 Lecture 32 241 a. Variations on a theme: Other horseshoes 241 b. Intermittent chaos vs. persistent chaos 245 с Homoclinic orbits and horseshoes 247 Chapter 9. Continuous-Time Systems: The Lorenz Model 253 Lecture 33 253 a. Continuous-time systems: Basic concepts 253 b. Fixed points of continuous-time systems 256 Lecture 34 259 a. The pendulum 259 b. Two-dimensional systems 263 Lecture 35 267 a. The Lorenz equations 267 b. Beyond the linear mindset 269 с Examining the Lorenz system 269 Lecture 36 274 a. Passing to a Poincaré map 274 b. Horseshoes in the Lorenz system 277 Lecture 37 281 a. The Lorenz attractor 281 b. The geometric Lorenz attractor 281 с Dimension of the geometric Lorenz attractor 285 d. Back to the Lorenz attractor, and beyond 286 Appendix 289 Hints to selected exercises 295 Suggested Reading 299 Bibliography 305 Index 309
any_adam_object	1
author	Pesin, Yakov B. 1946-
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isbn	9780821848890
language	English
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physical	XVI, 314 S. Ill., graph. Darst.
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publisher	American Math. Soc. [u.a.]
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spelling	Pesin, Yakov B. 1946- Verfasser (DE-588)129370533 aut Lectures on fractal geometry and dynamical systems Yakov Pesin ; Vaughn Climenhaga Providence, RI American Math. Soc. [u.a.] 2009 XVI, 314 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Student mathematical library 52 Includes bibliographical references and index Fractals Dynamics Climenhaga, Vaughn 1982- Sonstige (DE-588)139069801 oth Student mathematical library 52 (DE-604)BV013184751 52 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018740200&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Pesin, Yakov B. 1946- Lectures on fractal geometry and dynamical systems Student mathematical library Fractals Dynamics
title	Lectures on fractal geometry and dynamical systems
title_auth	Lectures on fractal geometry and dynamical systems
title_exact_search	Lectures on fractal geometry and dynamical systems
title_full	Lectures on fractal geometry and dynamical systems Yakov Pesin ; Vaughn Climenhaga
title_fullStr	Lectures on fractal geometry and dynamical systems Yakov Pesin ; Vaughn Climenhaga
title_full_unstemmed	Lectures on fractal geometry and dynamical systems Yakov Pesin ; Vaughn Climenhaga
title_short	Lectures on fractal geometry and dynamical systems
title_sort	lectures on fractal geometry and dynamical systems
topic	Fractals Dynamics
topic_facet	Fractals Dynamics
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018740200&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV013184751
work_keys_str_mv	AT pesinyakovb lecturesonfractalgeometryanddynamicalsystems AT climenhagavaughn lecturesonfractalgeometryanddynamicalsystems

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