An introduction to both topics in dynamical systems and mathematical thinking. In particular, the authors emphasize those parts of mathematical analysis necessary for understanding the intricacies of a discrete dynamical system. The organizing principle is the understanding of the parametrized family of functions h(x) = rx(1-x). Readers should have some background in calculus although extensive knowledge of proof-based mathematics is not necessary. Students will learn to understand periodic points, stable sets, bifurcations, symbolic dynamics, and chaos. The book includes rigorous proofs of important concepts in dynamics while remaining accessible to the typical advanced undergraduate.
Reihe
Auflage
Softcover reprint of the original 1st ed. 1994
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Produkt-Hinweis
Illustrationen
black & white illustrations
Maße
Höhe: 23.5 cm
Breite: 15.5 cm
Dicke: 12 mm
Gewicht
ISBN-13
978-0-387-94208-7 (9780387942087)
DOI
10.1007/978-1-4684-0222-3
Schweitzer Klassifikation
1 Introduction.- 1.1 Phase Portraits.- Exercise Set 1.- 2 A Quick Look at Functions.- Exercise Set 2.- 3 The Topology of the Real Numbers.- Exercise Set 3.- 4 Periodic Points and Stable Sets.- 4.1 Graphical Analysis.- Exercise Set 4.- 5 Sarkovskii's Theorem.- Exercise Set 5.- 6 Differentiability and Its Implications.- Exercise Set 6.- 7 Parametrized Families of Functions and Bifurcations.- Exercise Set 7.- 8 The Logistic Function, Part I.- Exercise Set 8.- 9 Symbolic Dynamics and Chaos.- Exercise Set 9.- 10 The Logistic Function, Part II: Topological Conjugacy.- Exercise Set 10.- 11 The Logistic Function, Part III.- Exercise Set 11.- 12 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.- 13 Numerical Solutions of Differential Equations.- Exercise Set 13.- 14 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.- 15 The Quadratic Family and the Mandelbrot Set.- Generating Julia and Mandelbrot Sets on a Computer.- Exercise Set 15.- Appendix A. Computer Algorithms.- A.1 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.- Topics in Mathematics.- General Interest Books on Dynamics.- Computer Programs and Algorithms.- List of Symbols.
