A Modern Introduction to Dynamical Systems 1st Edition by Richard Brown ISBN 9780198743286 0198743289

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ISBN 10: 0198743289
ISBN 13: 9780198743286
Author: Richard Brown

A Modern Introduction to Dynamical Systems 1st Edition Table of contents:

1 What Is a Dynamical System?

1.1 Definitions

1.1.1 Ordinary Differential Equations (ODEs)

1.1.2 Maps

1.1.3 Symbolic Dynamics

1.1.4 Billiards

1.1.5 Higher-Order Recursions

1.2 The Viewpoint

2 Simple Dynamics

2.1 Preliminaries

2.1.1 A Simple System

2.1.2 The Time-t Map

2.1.3 Metrics on Sets

2.1.4 Lipschitz Continuity

2.2 The Contraction Principle

2.2.1 Contractions on Intervals

2.2.2 Contractions in Several Variables

2.2.3 Application: The Newton–Raphson Method

2.2.4 Application: Existence and Uniqueness of ODE Solutions

2.2.5 Application: Heron of Alexandria

2.3 Interval Maps

2.3.1 Cobwebbing

2.3.2 Fixed-Point Stability

2.3.3 Monotonic Maps

2.3.4 Homoclinic/Heteroclinic Points

2.4 Bifurcations of Interval Maps

2.4.1 Saddle-Node Bifurcation

2.4.2 Transcritical Bifurcation

2.4.3 Pitchfork Bifurcation

2.5 First Return Maps

2.6 A Quadratic Interval Map: The Logistic Map

3 The Objects of Dynamics

3.1 Topology on Sets

3.2 More on Metrics

3.2.1 More on Lipschitz Continuity

3.2.2 Metric Equivalence

3.2.3 Fixed-Point Theorems

3.3 Some Non-Euclidean Metric Spaces

3.3.1 The n-Sphere

3.3.2 The Unit Circle

3.3.3 The Cylinder

3.3.4 The 2-Torus

3.4 A Cantor Set

3.4.1 The Koch Curve

3.4.2 Sierpinski Carpet

3.4.3 The Sponges

4 Flows and Maps of Euclidean Space

4.1 Linear, First-order ODE Systems in the Plane

4.1.1 General Homogeneous, Linear Systems in Euclidean Space

4.1.2 Autonomous Linear Systems

4.1.3 The Matrix Exponential

4.1.4 Two-Dimensional Classification

4.2 Bifurcations in Linear Planar Systems

4.2.1 Linearized Poincaré–Andronov–Hopf Bifurcation

4.2.2 Saddle-Node Bifurcation

4.3 Linear Planar Maps

4.3.1 Nodes: Sinks and Sources

4.3.2 Star or Proper Nodes

4.3.3 Degenerate or Improper Nodes

4.3.4 Spirals and Centers

4.3.5 Saddle Points

4.4 Linear Flows versus Linear Maps

4.5 Local Linearization and Stability of Equilibria

4.6 Isolated Periodic Orbit Stability

4.6.1 The Poincaré–Bendixson Theorem

4.6.2 Limit Sets of Flows

4.6.3 Flows in the Plane

4.6.4 Application: The van der Pol Oscillator

4.6.5 The Poincaré–Andronov–Hopf Bifurcation

4.7 Application: Competing Species

4.7.1 The Fixed Points

4.7.2 Type and Stability

5 Recurrence

5.1 Rotations of the circle

5.1.1 Continued Fraction Representation

5.2 Equidistribution and Weyl’s Theorem

5.2.1 Application: Periodic Function Reconstruction via Sampling

5.3 Linear Flows on the Torus

5.3.1 Application: Lissajous Figures

5.3.2 Application: A Polygonal Billiard

5.4 Toral Translations

5.5 Invertible Circle Maps

6 Phase Volume Preservation

6.1 Incompressibility

6.2 Newtonian Systems of Classical Mechanics

6.2.1 Generating Flows from Functions: Lagrange

6.2.2 Generating Flows from Functions: Hamilton

6.2.3 Exact Differential Equations

6.2.4 Application: The Planar Pendulum

6.2.5 First Integrals

6.2.6 Application: The Spherical Pendulum

6.3 Poincaré Recurrence

6.3.1 Non-Wandering Points

6.3.2 The Poincaré Recurrence Theorem

6.4 Billiards

6.4.1 Circular Billiards

6.4.2 Elliptic Billiards

6.4.3 General Convex Billiards

6.4.4 Poincaré’s Last Geometric Theorem

6.4.5 Application: Pitcher Problems

7 Complicated Orbit Structure

7.1 Counting Periodic Orbits

7.1.1 The Quadratic Map: Beyond 4

7.1.2 Hyperbolic Toral Automorphisms

7.1.3 Application: Image Restoration

7.1.4 Inverse Limit Spaces

7.1.5 Shift Spaces

7.1.6 Markov Partitions

7.1.7 Application: The Baker’s Transformation

7.2 Two-Dimensional Markov Partitions: Arnol’d’s Cat Map

7.3 Chaos and Mixing

7.4 Sensitive Dependence on Initial Conditions

7.5 Quadratic Maps: The Final Interval

7.5.1 Period-Doubling Bifurcation

7.5.2 The Schwarzian Derivative

7.5.3 Sharkovskii’s Theorem

7.6 Two More Examples of Complicated Dynamical Systems

7.6.1 Complex Dynamics

7.6.2 Smale Horseshoe

8 Dynamical Invariants

8.1 Topological Conjugacy

8.1.1 Conjugate Maps

8.1.2 Conjugate Flows

8.1.3 Conjugacy as Classification

8.2 Topological Entropy

8.2.1 Lyapunov Exponents

8.2.2 Capacity

8.2.3 Box Dimension

8.2.4 Bowen–Dinaburg (Metric) Topological Entropy

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