Chaotic Maps

This book is developed from the lecture notes on dynamical systems and chaos the two authors taught at the Mathematics Departments of Texas A&M University and Zhongshan (Sun Yat-Sen) University in Guangzhou, China during 1995-2011. The materials in the notes are intended for a semester-long introductory course. The main objective is to familiarize the students with the theory and techniques for (discrete-time) maps from mainly an analysis viewpoint, aiming eventually to also provide a stepping stone for nonlinear systems governed by ODEs and PDEs. The book is divided into ten chapters and two appendices. They cover the following major themes: (I) Interval maps: Their basic properties (Chapter 1), Sharkovski's Theorem on periodicities (Chapter 3), bifurcations (Chapter 4), and homoclinicity (Chapter 5). (II) General dynamical systems and Smale Horseshoe: The 2- and ksymbol dynamics, topological conjugacy and shift invariant sets (Chapter 6), and the Smale Horseshoe (Chapter 7). (III) Rapid fluctuations and fractals: Total variations and heuristics (Chapter 2), fractals (Chapter 8), and rapid fluctuations of multi-dimensional maps and infinite-dimensional maps (Chapters 9 and 10).