Dynamical Systems Modeling and Analysis

9.

9.1.

Foundations of Chaos

9.1.1.

Defining Characteristics

9.1.1.1.

Sensitive Dependence on Initial Conditions

9.1.1.2.

Topological Mixing

9.1.1.3.

Dense Periodic Orbits

9.1.2.

Mathematical Definitions

9.1.2.1.

Devaney's Definition

9.1.2.2.

9.1.3.

Historical Development

9.2.

Routes to Chaos

9.2.1.

Period-Doubling Route

9.2.1.1.

Cascade Structure

9.2.1.2.

Feigenbaum Constants

9.2.1.3.

9.2.2.

Intermittency Route

9.2.2.1.

Type I Intermittency

9.2.2.2.

Type II Intermittency

9.2.2.3.

Type III Intermittency

9.2.3.

Quasiperiodic Route

9.2.3.1.

Torus Breakdown

9.2.3.2.

9.3.

9.3.1.

The Logistic Map

9.3.1.1.

Parameter Regimes

9.3.1.2.

Bifurcation Diagram

9.3.1.3.

Symbolic Dynamics

9.3.2.

9.3.2.1.

Strange Attractor

9.3.2.2.

Fractal Structure

9.3.3.

The Baker's Map

9.3.3.1.

Stretching and Folding

9.3.3.2.

Symbolic Representation

9.4.

Strange Attractors

9.4.1.

Definition and Properties

9.4.2.

The Lorenz Attractor

9.4.2.1.

Lorenz Equations

9.4.2.2.

Geometric Structure

9.4.2.3.

Butterfly Effect

9.4.3.

The Rössler Attractor

9.4.3.1.

System Equations

9.4.3.2.

Spiral Structure

9.5.

Characterizing Chaos

9.5.1.

Lyapunov Exponents

9.5.1.1.

Definition and Calculation

9.5.1.2.

Spectrum of Exponents

9.5.1.3.

9.5.2.

Fractal Dimensions

9.5.2.1.

Hausdorff Dimension

9.5.2.2.

Box-Counting Dimension

9.5.2.3.

Correlation Dimension

9.5.3.

Entropy Measures

9.5.3.1.

Topological Entropy

9.5.3.2.

9.5.4.

Poincaré Sections

9.5.4.1.

Construction Methods

9.5.4.2.

Dimensional Reduction

9.5.4.3.

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8. Bifurcation Theory

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10. Numerical Methods and Computational Analysis