A First Course in Discrete Dynamical Systems

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.- Exercise Set 4.- 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.- Exercise Set 8.- 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.- Exercise Set 11.- 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.- Exercise Set 12.- 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.- Exercise Set 14.- The Quadratic Family and the Mandelbrot Set.- 15.1 Generating Julia and Mandelbrot Sets on a Computer.- Exercise Set 15.- 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.