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This book was developed from lecture notes for an introductory graduate course and provides an essential introduction to chaotic maps in finite-dimensional spaces. Furthermore, the authors show how to apply this theory to infinite-dimensional systems corresponding to partial differential equations to study chaotic vibration of the wave equation subject to various types of nonlinear boundary conditions. The book provides background on chaos as a highly interesting nonlinear phenomenon and explains why it is one of the most important scientific findings of the past three decades. In addition, the book covers key topics including one-dimensional dynamical systems, bifurcations, general topological, symbolic dynamical systems, and fractals. The authors also show a class of infinite-dimensional nonlinear dynamical systems, which are reducible to interval maps, plus rapid fluctuations of chaotic maps. This second edition includes updated and expanded chapters as well as additional problems.
List 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.
About the author
Liangliang Li, Ph.D., is an Associate Professor at Sun Yat-Sen University. He received his B.S. from China Three Gorges University and his Ph.D. in Mathematics from Sun Yat-Sen University. His research focuses on investigating new electronic materials and devices.
Yu Huang, Ph.D., is a Professor at Sun Yat-Sen University’s School of Science. He received his B.S. and M.S. from Sun Yat-Sen University and his Ph.D. from the Chinese University of Hong Kong. Dr. Huang is an Associate Editor of the Journal of Mathematical Analysis and Applications. His research interests include dynamical system theory, ergodic theory, switched systems, and networked control systems.
Goong Chen, Ph.D., is a 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.
Summary
This book was developed from lecture notes for an introductory graduate course and provides an essential introduction to chaotic maps in finite-dimensional spaces. Furthermore, the authors show how to apply this theory to infinite-dimensional systems corresponding to partial differential equations to study chaotic vibration of the wave equation subject to various types of nonlinear boundary conditions. The book provides background on chaos as a highly interesting nonlinear phenomenon and explains why it is one of the most important scientific findings of the past three decades. In addition, the book covers key topics including one-dimensional dynamical systems, bifurcations, general topological, symbolic dynamical systems, and fractals. The authors also show a class of infinite-dimensional nonlinear dynamical systems, which are reducible to interval maps, plus rapid fluctuations of chaotic maps. This second edition includes updated and expanded chapters as well as additional problems.
