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Chaotic maps dynamics, fractals, and rapid fluctuations
Author
Title
Chaotic maps [electronic resource] : dynamics, fractals, and rapid fluctuations / Goong Chen and Yu Huang.
ISBN
9781598299151 (electronic bk.)
9781598299144 (pbk.)
Published
San Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA) : Morgan & Claypool, c2011.
Physical Description
1 online resource (xiii, 227 p.) : ill., digital file.
Local Notes
Access is available to the Yale community.
Notes
Part of: Synthesis digital library of engineering and computer science.
Series from website.
Access and use
Access restricted by licensing agreement.
Summary
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.
Other formats
Also available in print.
Format
Books / Online
Language
English
Added to Catalog
April 24, 2013
Series
Synthesis lectures on mathematics and statistics, # 11
System details note
Mode of access: World Wide Web.
System requirements: Adobe Acrobat Reader.
Bibliography
Includes bibliographical references (p. 217-222) and index.
Contents
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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