Richard A. Holmgren

A First Course in Discrete Dynamical Systems (eBook, PDF)

• Format: PDF

Produktbeschreibung

Given the ease with which computers can do iteration it is now possible for almost anyone 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. Mathematica programs that illustrate the dynamics are included in an appendix.

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Produktdetails

• Verlag: Springer US
• Seitenzahl: 223
• Erscheinungstermin: 5. September 2012
• Englisch
• ISBN-13: 9781441987327
• Artikelnr.: 44181639

Inhaltsangabe

1.1. Phase Portraits.- Exercise Set 1.- A Quick Look at Functions.- Exercise Set 2.- The Topology of the Real Numbers.- Exercise Set 3.- Periodic Points and Stable Sets.- 4.1 Graphical Analysis.- Sarkovskii's Theorem.- Exercise Set 5.- Differentiability and Its Implications.- Exercise Set 6.- Parametrized Families of Functions and Bifurcations.- Exercise Set 7.- The Logistic Function Part I: Cantor Sets and Chaos.- 8.1. A First Look at the Logistic Function when r > 4.- 8.2. Cantor Sets.- 8.3. Chaos and the Dynamics of the Logistic Function.- 8.4. A Few Additional Comments on Cantor Sets.- The Logistic Function Part II: Topological Conjugacy.- Exercise Set 9.- The Logistic Function Part III: A Period-Doubling Cascade.- Exercise Set 10.- The Logistic Function Part IV: Symbolic Dynamics.- 11.1. Symbolic Dynamics and Metric Spaces.- 11.2. Symbolic Dynamics and the Logistic Function.- Newton's Method.- 12.1 Newton's Method for Quadratic Functions.- 12.2 Newton's Method for Cubic Functions.- 12.3 Intervals and Rates of Convergence.- Numerical Solutions of Differential Equations.- Exercise Set 13.- The Dynamics of Complex Functions.- 14.1. The Complex Numbers.- 14.2. Complex Functions.- 14.3. The Dynamics of Complex Functions.- 14.4. The Riemann Sphere.- 14.5. Newton's Method in the Complex Plane.- The Quadratic Family and the Mandelbrot Set.- 15.1 Generating Julia and Mandelbrot Sets on a Computer.- A.l. Iterating Functions.- Finding the Value of a Point Under Iteration.- Tables of Iterates.- Controlling the Precision of the Computations.- Graphing Iterated Functions.- A.2. Graphical Analysis.- A.3. Bifurcation Diagrams.- A.4. Julia Sets.- A.5 The Mandelbrot Set.- A.6 Stable Sets of Newton's Method.- References.- Dynamical Systems.- General Interest Books on Dynamics.-Topics in Mathematics.- Computer Programs and Algorithms.