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An Introduction To Chaotic Dynamical Systems 3rd Edition by Robert L Devaney ISBN 1032150467 9781032150468

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Author(s): Robert L. Devaney
ISBN(s): 9781032150468, 1032150467
Edition: 3
File Details: PDF, 111.24 MB
Year: 2021
Language: english
SKU: EB-42318316 Category: Tag:

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ISBN 10: 1032150467 
ISBN 13: 9781032150468
Author: Robert L Devaney

There is an explosion of interest in dynamical systems in the mathematical community as well as in many areas of science. The results have been truly exciting: systems which once seemed completely intractable from an analytic point of view can now be understood in a geometric or qualitative sense rather easily. Scientists and engineers realize the power and the beauty of the geometric and qualitative techniques. These techniques apply to a number of important nonlinear problems ranging from physics and chemistry to ecology and economics. Computer graphics have allowed us to view the dynamical behavior geometrically. The appearance of incredibly beautiful and intricate objects such as the Mandelbrot set, the Julia set, and other fractals have really piqued interest in the field. This is text is aimed primarily at advanced undergraduate and beginning graduate students. Throughout, the author emphasizes the mathematical aspects of the theory of discrete dynamical systems, not the many and diverse applications of this theory. The field of dynamical systems and especially the study of chaotic systems has been hailed as one of the important breakthroughs in science in the past century and its importance continues to expand. There is no question that the field is becoming more and more important in a variety of scientific disciplines. New to this edition: •Greatly expanded coverage complex dynamics now in Chapter 2 •The third chapter is now devoted to higher dimensional dynamical systems. •Chapters 2 and 3 are independent of one another. •New exercises have been added throughout.

An Introduction To Chaotic Dynamical Systems 3rd Table of contents:

I One Dimensional Dynamics
1 A Visual and Historical Tour

  • 1.1 Images from Dynamical Systems

  • 1.2 A Brief History of Dynamics

2 Examples of Dynamical Systems

  • 2.1 Population Models

  • 2.2 Newton’s Method

3 Elementary Definitions

  • 3.1 Orbits

  • 3.2 Geometric Views of Orbits

4 Hyperbolicity

  • 4.1 Types of Periodic Points

  • 4.2 A Glimpse of Bifurcations

5 An Example: The Logistic Family

  • 5.1 The Simplest Case

  • 5.2 The Cantor Set Case

6 Symbolic Dynamics

  • 6.1 The Sequence Space

  • 6.2 The Shift Map

7 Topological Conjugacy

  • 7.1 The Itinerary Map

  • 7.2 Conjugacy

8 Chaos

9 Structural Stability

10 Sharkovsky’s Theorem

11 The Schwarzian Derivative

12 Bifurcations

  • 12.1 Examples of Bifurcations

  • 12.2 General Bifurcation Theorems

13 Another View of Period Three

  • 13.1 Subshifts of Finite Type

  • 13.2 The Period 3 Case

14 The Period-Doubling Route to Chaos

  • 14.1 Renormalization

  • 14.2 The Orbit Diagram

15 Homoclinic Points and Bifurcations

  • 15.1 Homoclinic Points

  • 15.2 Homoclinic Bifurcations

16 Maps of the Circle

  • 16.1 Rotation Numbers

  • 16.2 The Standard Family

17 Morse-Smale Diffeomorphisms

II Complex Dynamics
18 Quadratic Maps Revisited

  • 18.1 The Case c = 0

  • 18.2 The Case |c| > 2

  • 18.3 The Case c = −2

19 Normal Families and Exceptional Points

20 Periodic Points

  • 20.1 Linearization

  • 20.2 Critical Values in the Basins of Attraction

21 Properties of the Julia Set

22 The Geometry of the Julia Sets

  • 22.1 Quadratic Julia Sets

  • 22.2 A Julia Set for a Rational Map

  • 22.3 Fractals

23 Neutral Periodic Points

  • 23.1 Rationally Indifferent Periodic Points

  • 23.2 Irrationally Indifferent Periodic Points

24 The Mandelbrot Set

  • 24.1 Connectivity of the Julia Set

  • 24.2 The Mandelbrot Set

  • 24.3 Complex Bifurcations

  • 24.4 Geometry of the Principal Bulbs

  • 24.5 External Rays in the Dynamical Plane

  • 24.6 External Rays in the Parameter Plane

25 Rational Maps

  • 25.1 Singular Perturbations

  • 25.2 Basic Properties

  • 25.3 The Escape Trichotomy

  • 25.4 The Special Case n = 2

  • 25.5 Sierpinski Holes

26 The Exponential Family

  • 26.1 The Cantor Bouquet Case

  • 26.2 The Julia Set of e^z

  • 26.3 Indecomposable Continua

III Higher Dimensional Dynamics
27 Dynamics of Linear Maps

  • 27.1 Behavior of Linear Maps

  • 27.2 Stable and Unstable Subspaces

28 The Smale Horseshoe Map

  • 28.1 Symbolic Dynamics

29 Hyperbolic Toral Automorphisms

  • 29.1 Hyperbolic Toral Automorphisms

  • 29.2 Markov Partitions

30 Attractors

  • 30.1 The Solenoid

  • 30.2 The Plykin Attractor

31 The Stable and Unstable Manifold Theorem

32 Global Results and Hyperbolic Maps

33 The Hopf Bifurcation

  • 33.1 Planar Bifurcations

  • 33.2 Normal Forms

  • 33.3 The Hopf Bifurcation Theorem

34 The Hénon Map

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Tags: Robert L Devaney, Chaotic Dynamical, Systems