This book consists of lecture notes for a semester-long introductory graduate course on dynamical systems and chaos taught by the authors at Texas A&M University and Zhongshan University, China. There are ten chapters in the main body of the book, covering an elementary theory of chaotic maps in finite-dimensional spaces. The topics include one-dimensional dynamical systems (interval maps), bifurcations, general topological, symbolic dynamical systems, fractals and a class of infinite-dimensional dynamical systems which are induced by interval maps, plus rapid fluctuations of chaotic maps as a new viewpoint developed by the authors in recent years. Two appendices are also provided in order to ease the transitions for the readership from discrete-time dynamical systems to continuous-time dynamical systems, governed by ordinary and partial differential equations. Table of Contents: Simple Interval Maps and Their Iterations / Total Variations of Iterates of Maps / Ordering among Periods: The Sharkovski Theorem / Bifurcation Theorems for Maps / Homoclinicity. Lyapunoff Exponents / Symbolic Dynamics, Conjugacy and Shift Invariant Sets / The Smale Horseshoe / Fractals / Rapid Fluctuations of Chaotic Maps on RN / Infinite-dimensional Systems Induced by Continuous-Time Difference Equations

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Goong Chen was born in Kaohsiung, Taiwan in 1950. He received his BSc (Math) from the National Tsing Hua University in Hsinchu, Taiwan in 1972 and PhD (Math) from the University of Wisconsin at Madison in 1977. He has taught at the Southern Illinois University at Carbondale (1977-78), and the Pennsylvania State University at University Park (1978-1987). Since 1987, he has been Professor of Mathematics and Aerospace Engineering, and (since 2000) a member of the Institute for Quantum Science and Engineering, at Texas A&M University in College Station, Texas. Since 2010, he is also Professor of Mathematics at Texas A&M University in Qatar at Doha, Qatar. He has held visiting positions at INRIA in Rocquencourt, France, Centre de Recherche Mathematiques of the Universite de Montreal, the Technical University of Denmark in Lyngby, Denmark, the National University of Singapore, National Taiwan University in Taipei, Taiwan, Academia Sinica in Nankang, Taiwan, and National Tsing Hua University in Hsinchu, Taiwan. He has research interests in many areas of applied and computational mathematics: control theory for partial differential equations (PDEs), boundary element methods and numerical solutions of PDEs, engineering mechanics, chaotic dynamics, quantum computation, chemical physics and quantum mechanics. He has written over one hundred and thirty papers, five advanced texts/monographs, and co-edited four books. He is Editor-in-Chief of the Journal of Mathematical Analysis and Applications, and he has served on several other editorial boards, including the SIAM Journal on Control and Optimization, the International Journal on Quantum Information, and the Electronic Journal of Differential Equations. He is also a co-holder of a U.S. Patent on certain quantum circuit design for quantum computing. He holds memberships in the American Mathematical Society (AMS) and the Society for Industrial and Applied Mathematics (SIAM). Yu Huang was born in Guangdong Province, People's Republic of China, in 1963. He received his BSc and MSc (Math) from Zhongshan (Dr. Sun Yat-Sen) University in Guangzhou, China, respectively, in 1983 and 1986, and his PhD (Math) from the Chinese University of Hong Kong, in Hong Kong, in 1995. He has been teaching at Mathematics Department of Sun Yat-Sen University since 1986. There, he was promoted to Professor of Mathematics in 2006. His research interests include control theory for partial differential equations, topological dynamical systems and chaos, and switching system theory. He has written over forty papers and co-edited a book. He is an Associate Editor of the Journal of Mathematical Analysis and a guest Associate Editor of the International Journal of Bifurcation and Chaos.