First course in chaotic dynamical systems : theory and experiment / Robert L. Devaney, A
Chaotic behavior in systems Differentiable dynamical systems Mathematics sähkökirjat
Westview Press
1992
EISBN 9780813345475
Preface; Contents; Chapter 1 A Mathematical and Historical Tour; 1.1 Images from Dynamical Systems; 1.2 A Brief History of Dynamics; Chapter 2 Examples of Dynamical Systems; 2.1 An Example from Finance; 2.2 An Example from Ecology; 2.3 Finding Roots and Solving Equations; 2.4 Differential Equations; Chapter 3 Orbits; 3.1 Iteration; 3.2 Orbits; 3.3 Types of Orbits; 3.4 Other Orbits; 3.5 The Doubling Function; 3.6 Experiment: The Computer May Lie; Chapter 4 Graphical Analysis; 4.1 Graphical Analysis; 4.2 Orbit Analysis; 4.3 The Phase Portrait; Chapter 5 Fixed and Periodic Points
13.2 Convergence and NonconvergenceChapter 14 Fractals; 14.1 The Chaos Game; 14.2 The Cantor Set Revisited; 14.3 The Sierpinski Triangle; 14.4 The Koch Snowflake; 14.5 Topological Dimension; 14.6 Fractal Dimension; 14.7 Iterated Function Systems; Chapter 15 Complex Functions; 15.1 Complex Arithmetic; 15.2 Complex Square Roots; 15.3 Linear Complex Functions; 15.4 Calculus of Complex Functions; Chapter 16 The Julia Set; 16.1 The Squaring Function; 16.2 The Chaotic Quadratic Function; 16.3 Cantor Sets Again; 16.4 Computing the Filled Julia Set
16.5 Experiment: Filled Julia Sets and Critical Orbits16.6 The Julia Set as a Repellor; Chapter 17 The Mandelbrot Set; 17.1 The Fundamental Dichotomy; 17.2 The Mandelbrot Set; 17.3 Experiment: Periods of Other Bulbs; 17.4 Experiment: Periods of the Decorations; 17.5 Experiment: Find the Julia Set; 17.6 Experiment: Spokes and Antennas; 17.7 Experiment: Similarity of the Mandelbrot and Julia Sets; Chapter 18 Further Projects and Experiments; 18.1 The Tricorn; 18.2 Cubics; 18.3 Exponential Functions; 18.4 Trigonometric Functions; 18.5 Complex Newton's Method
5.1 A Fixed Point Theorem5.2 Attraction and Repulsion; 5.3 Calculus of Fixed Points; 5.4 Why is this true?; 5.5 Periodic Points; 5.6 Experiment: Rates of Convergence; Chapter 6 Bifurcations; 6.1 Dynamics of the Quadratic Map; 6.2 The Saddle-Node Bifurcation; 6.3 The Period-Doubling Bifurcation; 6.4 Experiment: The Transition to Chaos; Chapter 7 The Quadratic Family; 7.1 The Case c= -2; 7.2 The Case c < -2; 7.3 The Cantor Middle-Thirds Set; Chapter 8 Transition to Chaos; 8.1 The Orbit Diagram; 8.2 The Period-Doubling Route to Chaos; 8.3 Experiment: Windos in the Orbit Diagram
Chapter 9 Symbolic Dynamics9.1 Itineraries; 9.2 The Sequence Space; 9.3 The Shift Map; 9.4 Conjugacy; Chapter 10 Chaos; 10.1 Three Properties of a Chaotic System; 10.2 Other Chaotic Systems; 10.3 Manifestations of Chaos; 10.4 Experiment: Feigenbaum's Constant; Chapter 11 Sarkovskii's Theorem; 11.1 Period 3 Implies Chaos; 11.2 Sarkovskii's Theorem; 11.3 The Period-3 Window; 11.4 Subshifts of Finite Type; Chapter 12 The Role of the Critical Orbit; 12.1 The Schwarzian Derivative; 12.2 The Critical Point and Basins of Attraction; Chapter 13 Newton's Method; 13.1 Basic Properties
This is the first book to introduce modern topics in dynamical systems at the undergraduate level. Accessible to readers with only a background in calculus, the book integrates both theory and computer experiments into its coverage of contemporary ideas in dynamics. It is designed as a gradual introduction to the basic mathematical ideas behind such topics as chaos, fractals, Newton's method, symbolic dynamics, the Julia set, and the Mandelbrot set, and includes biographies of some of the leading researchers in the field of dynamical systems. Mathematical and computer experiments are integrate.
13.2 Convergence and NonconvergenceChapter 14 Fractals; 14.1 The Chaos Game; 14.2 The Cantor Set Revisited; 14.3 The Sierpinski Triangle; 14.4 The Koch Snowflake; 14.5 Topological Dimension; 14.6 Fractal Dimension; 14.7 Iterated Function Systems; Chapter 15 Complex Functions; 15.1 Complex Arithmetic; 15.2 Complex Square Roots; 15.3 Linear Complex Functions; 15.4 Calculus of Complex Functions; Chapter 16 The Julia Set; 16.1 The Squaring Function; 16.2 The Chaotic Quadratic Function; 16.3 Cantor Sets Again; 16.4 Computing the Filled Julia Set
16.5 Experiment: Filled Julia Sets and Critical Orbits16.6 The Julia Set as a Repellor; Chapter 17 The Mandelbrot Set; 17.1 The Fundamental Dichotomy; 17.2 The Mandelbrot Set; 17.3 Experiment: Periods of Other Bulbs; 17.4 Experiment: Periods of the Decorations; 17.5 Experiment: Find the Julia Set; 17.6 Experiment: Spokes and Antennas; 17.7 Experiment: Similarity of the Mandelbrot and Julia Sets; Chapter 18 Further Projects and Experiments; 18.1 The Tricorn; 18.2 Cubics; 18.3 Exponential Functions; 18.4 Trigonometric Functions; 18.5 Complex Newton's Method
5.1 A Fixed Point Theorem5.2 Attraction and Repulsion; 5.3 Calculus of Fixed Points; 5.4 Why is this true?; 5.5 Periodic Points; 5.6 Experiment: Rates of Convergence; Chapter 6 Bifurcations; 6.1 Dynamics of the Quadratic Map; 6.2 The Saddle-Node Bifurcation; 6.3 The Period-Doubling Bifurcation; 6.4 Experiment: The Transition to Chaos; Chapter 7 The Quadratic Family; 7.1 The Case c= -2; 7.2 The Case c < -2; 7.3 The Cantor Middle-Thirds Set; Chapter 8 Transition to Chaos; 8.1 The Orbit Diagram; 8.2 The Period-Doubling Route to Chaos; 8.3 Experiment: Windos in the Orbit Diagram
Chapter 9 Symbolic Dynamics9.1 Itineraries; 9.2 The Sequence Space; 9.3 The Shift Map; 9.4 Conjugacy; Chapter 10 Chaos; 10.1 Three Properties of a Chaotic System; 10.2 Other Chaotic Systems; 10.3 Manifestations of Chaos; 10.4 Experiment: Feigenbaum's Constant; Chapter 11 Sarkovskii's Theorem; 11.1 Period 3 Implies Chaos; 11.2 Sarkovskii's Theorem; 11.3 The Period-3 Window; 11.4 Subshifts of Finite Type; Chapter 12 The Role of the Critical Orbit; 12.1 The Schwarzian Derivative; 12.2 The Critical Point and Basins of Attraction; Chapter 13 Newton's Method; 13.1 Basic Properties
This is the first book to introduce modern topics in dynamical systems at the undergraduate level. Accessible to readers with only a background in calculus, the book integrates both theory and computer experiments into its coverage of contemporary ideas in dynamics. It is designed as a gradual introduction to the basic mathematical ideas behind such topics as chaos, fractals, Newton's method, symbolic dynamics, the Julia set, and the Mandelbrot set, and includes biographies of some of the leading researchers in the field of dynamical systems. Mathematical and computer experiments are integrate.
