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Part of: Synthesis digital library of engineering and computer science.

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Includes bibliographical references (p. 217-222) and index.

1. Simple interval maps and their iterations -- 1.1 Introduction -- 1.2 The inverse and implicit function theorems -- 1.3 Visualizing from the graphics of iterations of the quadratic map -- Notes for chapter 1 --

2. Total variations of iterates of maps -- 2.1 The use of total variations as a measure of chaos -- Notes for chapter 2 --

3. Ordering among periods: the Sharkovski theorem -- Notes for chapter 3 --

4. Bifurcation theorems for maps -- 4.1 The period-doubling bifurcation theorem -- 4.2 Saddle-node bifurcations -- 4.3 The pitchfork bifurcation -- 4.4 Hopf bifurcation -- Notes for chapter 4 --

5. Homoclinicity. Lyapunoff exponents -- 5.1 Homoclinic orbits -- 5.2 Lyapunoff exponents -- Notes for chapter 5 --

6. Symbolic dynamics, conjugacy and shift invariant sets -- 6.1 The itinerary of an orbit -- 6.2 Properties of the shift map -- 6.3 Symbolic dynamical systems -- 6.4 The dynamics of [Sigma ...] and chaos -- 6.5 Topological conjugacy and semiconjugacy -- 6.6 Shift invariant sets -- 6.7 Construction of shift invariant sets -- 6.8 Snap-back repeller as a shift invariant set -- Notes for chapter 6 --

7. The Smale horseshoe -- 7.1 The standard Smale horseshoe -- 7.2 The general horseshoe -- Notes for chapter 7 --

8. Fractals -- 8.1 Examples of fractals -- 8.2 Hausdorff dimension and the Hausdorff measure -- 8.3 Iterated function systems (IFS) -- Notes for chapter 8 --

9. Rapid fluctuations of chaotic maps on RN -- 9.1 Total variation for vector-value maps -- 9.2 Rapid fluctuations of maps on RN -- 9.3 Rapid fluctuations of systems with quasi-shift invariant sets -- 9.4 Rapid fluctuations of systems containing topological horseshoes -- 9.5 Examples of applications of rapid fluctuations -- Notes for chapter 9 --

10. Infinite-dimensional systems induced by continuous-time difference equations -- 10.1 I3DS -- 10.2 Rates of growth of total variations of iterates -- 10.3 Properties of the set B(f ) -- 10.4 Properties of the set U(f ) -- 10.5 Properties of the set E(f ) -- Notes for chapter 10 --

A. Introduction to continuous-time dynamical systems -- The local behavior of 2-dimensional nonlinear systems -- Index for two-dimensional systems -- The Poincare map for a periodic orbit in RN -- B. Chaotic vibration of the wave equation due to energy pumping and van der Pol boundary conditions -- The mathematical model and motivations -- Chaotic vibration of the wave equation -- Authors' biographies -- Index.

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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.

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Title from PDF t.p. (viewed on September 25, 2011).