Softcover ISBN:  9780821848890 
Product Code:  STML/52 
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Individual Price:  $44.00 
Electronic ISBN:  9781470412210 
Product Code:  STML/52.E 
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Book DetailsStudent Mathematical LibraryVolume: 52; 2009; 314 ppMSC: Primary 37;
Both fractal geometry and dynamical systems have a long history of development and have provided fertile ground for many great mathematicians and much deep and important mathematics. These two areas interact with each other and with the theory of chaos in a fundamental way: many dynamical systems (even some very simple ones) produce fractal sets, which are in turn a source of irregular “chaotic” motions in the system. This book is an introduction to these two fields, with an emphasis on the relationship between them.
The first half of the book introduces some of the key ideas in fractal geometry and dimension theory—Cantor sets, Hausdorff dimension, box dimension—using dynamical notions whenever possible, particularly onedimensional Markov maps and symbolic dynamics. Various techniques for computing Hausdorff dimension are shown, leading to a discussion of Bernoulli and Markov measures and of the relationship between dimension, entropy, and Lyapunov exponents.
In the second half of the book some examples of dynamical systems are considered and various phenomena of chaotic behaviour are discussed, including bifurcations, hyperbolicity, attractors, horseshoes, and intermittent and persistent chaos. These phenomena are naturally revealed in the course of our study of two real models from science—the FitzHugh–Nagumo model and the Lorenz system of differential equations.
This book is accessible to undergraduate students and requires only standard knowledge in calculus, linear algebra, and differential equations. Elements of point set topology and measure theory are introduced as needed.
This book is a result of the MASS course in analysis at Penn State University in the fall semester of 2008.This book is published in cooperation with Mathematics Advanced Study Semesters.ReadershipUndergraduate and graduate students interested in dynamical systems and fractal geometry.

Table of Contents

Chapters

Chapter 1. Basic concepts and examples

Chapter 2. Fundamentals of dimension theory

Chapter 3. Measures: Definitions and examples

Chapter 4. Measures and dimensions

Chapter 5. Discretetime systems: The FitzHugh–Nagumo model

Chapter 6. The bifurcation diagram for the logistic map

Chapter 7. Chaotic attractors and persistent chaos

Chapter 8. Horseshoes and intermittent chaos

Chapter 9. Continuoustime systems: The Lorenz model

Appendix

Hints to selected exercises


Additional Material

Reviews

[F]or a student with a reasonable background in topology and measure theory this is a very useful book covering many of the main ideas in fractal geometry and dynamical systems in an accessible way, with a particular emphasis on dynamicallydefined fractals.
Ian Melbourne, Mathematical Reviews


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Both fractal geometry and dynamical systems have a long history of development and have provided fertile ground for many great mathematicians and much deep and important mathematics. These two areas interact with each other and with the theory of chaos in a fundamental way: many dynamical systems (even some very simple ones) produce fractal sets, which are in turn a source of irregular “chaotic” motions in the system. This book is an introduction to these two fields, with an emphasis on the relationship between them.
The first half of the book introduces some of the key ideas in fractal geometry and dimension theory—Cantor sets, Hausdorff dimension, box dimension—using dynamical notions whenever possible, particularly onedimensional Markov maps and symbolic dynamics. Various techniques for computing Hausdorff dimension are shown, leading to a discussion of Bernoulli and Markov measures and of the relationship between dimension, entropy, and Lyapunov exponents.
In the second half of the book some examples of dynamical systems are considered and various phenomena of chaotic behaviour are discussed, including bifurcations, hyperbolicity, attractors, horseshoes, and intermittent and persistent chaos. These phenomena are naturally revealed in the course of our study of two real models from science—the FitzHugh–Nagumo model and the Lorenz system of differential equations.
This book is accessible to undergraduate students and requires only standard knowledge in calculus, linear algebra, and differential equations. Elements of point set topology and measure theory are introduced as needed.
This book is a result of the MASS course in analysis at Penn State University in the fall semester of 2008.
Undergraduate and graduate students interested in dynamical systems and fractal geometry.

Chapters

Chapter 1. Basic concepts and examples

Chapter 2. Fundamentals of dimension theory

Chapter 3. Measures: Definitions and examples

Chapter 4. Measures and dimensions

Chapter 5. Discretetime systems: The FitzHugh–Nagumo model

Chapter 6. The bifurcation diagram for the logistic map

Chapter 7. Chaotic attractors and persistent chaos

Chapter 8. Horseshoes and intermittent chaos

Chapter 9. Continuoustime systems: The Lorenz model

Appendix

Hints to selected exercises

[F]or a student with a reasonable background in topology and measure theory this is a very useful book covering many of the main ideas in fractal geometry and dynamical systems in an accessible way, with a particular emphasis on dynamicallydefined fractals.
Ian Melbourne, Mathematical Reviews