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Systems Science
Dynamical Systems Modeling and Analysis
1. Introduction to Dynamical Systems
2. Mathematical Foundations
3. Modeling with Dynamical Systems
4. One-Dimensional Continuous Systems
5. One-Dimensional Discrete Systems
6. Two-Dimensional Linear Systems
7. Two-Dimensional Nonlinear Systems
8. Bifurcation Theory
9. Chaos Theory
10. Numerical Methods and Computational Analysis
11. Applications in Biology
12. Applications in Physics and Engineering
13. Applications in Chemistry
14. Applications in Economics and Social Sciences
15. Advanced Topics
4.
One-Dimensional Continuous Systems
4.1.
Basic Framework
4.1.1.
The Equation x' = f(x)
4.1.2.
Geometric Interpretation
4.1.3.
Solution Existence
4.2.
Fixed Points
4.2.1.
Definition and Calculation
4.2.2.
Graphical Identification
4.2.3.
Physical Interpretation
4.3.
Stability Analysis
4.3.1.
Linear Stability
4.3.1.1.
Derivative Test
4.3.1.2.
Stability Criteria
4.3.2.
Nonlinear Stability
4.3.2.1.
Lyapunov Stability
4.3.2.2.
Basin of Attraction
4.4.
Phase Line Analysis
4.4.1.
Construction of Phase Lines
4.4.2.
Flow Direction
4.4.3.
Equilibrium Classification
4.5.
Graphical Methods
4.5.1.
Plotting f(x) versus x
4.5.2.
Identifying Flow Patterns
4.5.3.
Sketching Solutions
4.6.
Potential Functions
4.6.1.
Definition and Construction
4.6.2.
Relationship to Dynamics
4.6.3.
Energy Interpretation
4.7.
Applications
4.7.1.
Population Growth Models
4.7.2.
Chemical Kinetics
4.7.3.
Mechanical Systems
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5. One-Dimensional Discrete Systems