Ebook: A First Course in Discrete Dynamical Systems

Discrete dynamical systems are essentially iterated functions. Given the ease with which computers can do iteration, it is now possible for anyone with access to a personal computer to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. The level of the presentation is suitable for advanced undergraduates with a year of calculus behind them. Students in the author's courses using this material have come from numerous disciplines; many have been majors in other disciplines who are taking mathematics courses out of general interest. Concepts from calculus are reviewed as necessary. Mathematica programs that illustrate the dynamics and that will aid the student in doing the exercises are included in an appendix.

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Discrete dynamical systems are essentially iterated functions. Given the ease with which computers can do iteration, it is now possible for anyone with access to a personal computer to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. The level of the presentation is suitable for advanced undergraduates with a year of calculus behind them. Students in the author's courses using this material have come from numerous disciplines; many have been majors in other disciplines who are taking mathematics courses out of general interest. Concepts from calculus are reviewed as necessary. Mathematica programs that illustrate the dynamics and that will aid the student in doing the exercises are included in an appendix.

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Discrete dynamical systems are essentially iterated functions. Given the ease with which computers can do iteration, it is now possible for anyone with access to a personal computer to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. The level of the presentation is suitable for advanced undergraduates with a year of calculus behind them. Students in the author's courses using this material have come from numerous disciplines; many have been majors in other disciplines who are taking mathematics courses out of general interest. Concepts from calculus are reviewed as necessary. Mathematica programs that illustrate the dynamics and that will aid the student in doing the exercises are included in an appendix.
Content:
Front Matter....Pages i-xv
Introduction....Pages 1-8
A Quick Look at Functions....Pages 9-20
The Topology of the Real Numbers....Pages 21-29
Periodic Points and Stable Sets....Pages 31-39
Sarkovskii’s Theorem....Pages 41-46
Differentiability and Its Implications....Pages 47-57
Parametrized Families of Functions and Bifurcations....Pages 59-67
The Logistic Function Part I: Cantor Sets and Chaos....Pages 69-86
The Logistic Function Part II: Topological Conjugacy....Pages 87-93
The Logistic Function Part III: A Period-Doubling Cascade....Pages 95-108
The Logistic Function Part IV: Symbolic Dynamics....Pages 109-126
Newton’s Method....Pages 127-151
Numerical Solutions of Differential Equations....Pages 153-165
The Dynamics of Complex Functions....Pages 167-192
The Quadratic Family and the Mandelbrot Set....Pages 193-202
Back Matter....Pages 203-224

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Discrete dynamical systems are essentially iterated functions. Given the ease with which computers can do iteration, it is now possible for anyone with access to a personal computer to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. The level of the presentation is suitable for advanced undergraduates with a year of calculus behind them. Students in the author's courses using this material have come from numerous disciplines; many have been majors in other disciplines who are taking mathematics courses out of general interest. Concepts from calculus are reviewed as necessary. Mathematica programs that illustrate the dynamics and that will aid the student in doing the exercises are included in an appendix.
Content:
Front Matter....Pages i-xv
Introduction....Pages 1-8
A Quick Look at Functions....Pages 9-20
The Topology of the Real Numbers....Pages 21-29
Periodic Points and Stable Sets....Pages 31-39
Sarkovskii’s Theorem....Pages 41-46
Differentiability and Its Implications....Pages 47-57
Parametrized Families of Functions and Bifurcations....Pages 59-67
The Logistic Function Part I: Cantor Sets and Chaos....Pages 69-86
The Logistic Function Part II: Topological Conjugacy....Pages 87-93
The Logistic Function Part III: A Period-Doubling Cascade....Pages 95-108
The Logistic Function Part IV: Symbolic Dynamics....Pages 109-126
Newton’s Method....Pages 127-151
Numerical Solutions of Differential Equations....Pages 153-165
The Dynamics of Complex Functions....Pages 167-192
The Quadratic Family and the Mandelbrot Set....Pages 193-202
Back Matter....Pages 203-224
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