Chaotic maps
Subtitle: dynamics, fractals, and rapid fluctuations
By: Goong Chen and Yu Huang
Contributors: Huang, Yu, 1963-
Series: Synthesis lectures on mathematics and statistics -- # 11
Publisher: Morgan & Claypool
Published: 2011
Place of publication: San Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA)
Language: eng
Format: [electronic resource] :
ISBN-13: 9781598299151; 9781598299144
Subjects: Chaotic behavior in systems; Mappings (Mathematics); Mathematics
Other titles: Synthesis digital library of engineering and computer science.
Description: 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.
Notes: Part of: Synthesis digital library of engineering and computer science.

Series from website.

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

Abstract freely available; full-text restricted to subscribers or individual document purchasers.

Also available in print.

Mode of access: World Wide Web.

System requirements: Adobe Acrobat Reader.
Table of contents: title: 1. Simple interval maps and their iterations; title: 1.1 Introduction; title: 1.2 The inverse and implicit function theorems; title: 1.3 Visualizing from the graphics of iterations of the quadratic map; title: Notes for chapter 1; title: 2. Total variations of iterates of maps; title: 2.1 The use of total variations as a measure of chaos; title: Notes for chapter 2; title: 3. Ordering among periods: the Sharkovski theorem; title: Notes for chapter 3; title: 4. Bifurcation theorems for maps; title: 4.1 The period-doubling bifurcation theorem; title: 4.2 Saddle-node bifurcations; title: 4.3 The pitchfork bifurcation; title: 4.4 Hopf bifurcation; title: Notes for chapter 4; title: 5. Homoclinicity. Lyapunoff exponents; title: 5.1 Homoclinic orbits; title: 5.2 Lyapunoff exponents; title: Notes for chapter 5; title: 6. Symbolic dynamics, conjugacy and shift invariant sets; title: 6.1 The itinerary of an orbit; title: 6.2 Properties of the shift map; title: 6.3 Symbolic dynamical systems; title: 6.4 The dynamics of [Sigma ...] and chaos; title: 6.5 Topological conjugacy and semiconjugacy; title: 6.6 Shift invariant sets; title: 6.7 Construction of shift invariant sets; title: 6.8 Snap-back repeller as a shift invariant set; title: Notes for chapter 6; title: 7. The Smale horseshoe; title: 7.1 The standard Smale horseshoe; title: 7.2 The general horseshoe; title: Notes for chapter 7; title: 8. Fractals; title: 8.1 Examples of fractals; title: 8.2 Hausdorff dimension and the Hausdorff measure; title: 8.3 Iterated function systems (IFS); title: Notes for chapter 8; title: 9. Rapid fluctuations of chaotic maps on RN; title: 9.1 Total variation for vector-value maps; title: 9.2 Rapid fluctuations of maps on RN; title: 9.3 Rapid fluctuations of systems with quasi-shift invariant sets; title: 9.4 Rapid fluctuations of systems containing topological horseshoes; title: 9.5 Examples of applications of rapid fluctuations; title: Notes for chapter 9; title: 10. Infinite-dimensional systems induced by continuous-time difference equations; title: 10.1 I3DS; title: 10.2 Rates of growth of total variations of iterates; title: 10.3 Properties of the set B(f ); title: 10.4 Properties of the set U(f ); title: 10.5 Properties of the set E(f ); title: Notes for chapter 10; title: A. Introduction to continuous-time dynamical systems; title: The local behavior of 2-dimensional nonlinear systems; title: Index for two-dimensional systems; title: The Poincare map for a periodic orbit in RN; title: B. Chaotic vibration of the wave equation due to energy pumping and van der Pol boundary conditions; title: The mathematical model and motivations; title: Chaotic vibration of the wave equation; title: Authors' biographies; title: Index.