Nonlinear Dynamics and Chaos

2.

One-Dimensional Flows

2.1.

Geometrical Analysis on the Line

2.1.1.

Flow Direction and Velocity

2.1.2.

Graphical Representation

2.1.3.

Time-to-Solution Curves

2.2.

Fixed Points and Equilibria

2.2.1.

Definition and Identification

2.2.2.

Graphical Methods for Finding Fixed Points

2.2.3.

Physical Interpretation

2.3.

Stability Analysis

2.3.1.

Linear Stability Theory

2.3.1.1.

Linearization Near Fixed Points

2.3.1.2.

Eigenvalue Analysis

2.3.1.3.

Stability Criteria

2.3.2.

Nonlinear Stability

2.3.2.1.

Lyapunov Stability

2.3.2.2.

Asymptotic Stability

2.3.2.3.

Exponential Stability

2.3.3.

Types of Fixed Points

2.3.3.1.

Stable (Attracting) Fixed Points

2.3.3.2.

Unstable (Repelling) Fixed Points

2.3.3.3.

Half-Stable (Semi-stable) Fixed Points

2.4.

Bifurcation Theory in 1D

2.4.1.

Bifurcation Fundamentals

2.4.1.1.

Parameter Dependence

2.4.1.2.

Qualitative Changes in Dynamics

2.4.1.3.

Bifurcation Points

2.4.2.

Bifurcation Diagrams

2.4.2.1.

Construction Methods

2.4.2.2.

Interpretation and Analysis

2.4.3.

Local Bifurcations

2.4.3.1.

Saddle-Node Bifurcation

2.4.3.2.

Transcritical Bifurcation

2.4.3.3.

Pitchfork Bifurcation

2.4.3.3.1.

Supercritical Pitchfork

2.4.3.3.2.

Subcritical Pitchfork

2.4.4.

2.4.4.1.

Canonical Forms for Bifurcations

2.4.4.2.

Unfolding Parameters

2.5.

Potential Functions

2.5.1.

Gradient Systems

2.5.2.

Mechanical Analogies

2.5.3.

Potential Wells and Barriers

2.5.4.

Energy Landscapes

2.5.5.

Relationship to Fixed Points

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1. Introduction to Dynamical Systems

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3. Two-Dimensional Flows