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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
5.
One-Dimensional Discrete Systems
5.1.
Basic Framework
5.1.1.
The Map x_{n+1} = f(x_n)
5.1.2.
Iteration Process
5.1.3.
Orbit Structure
5.2.
Fixed Points
5.2.1.
Definition and Calculation
5.2.2.
Graphical Identification
5.2.3.
Stability Analysis
5.2.3.1.
Linear Stability
5.2.3.2.
Derivative Criterion
5.3.
Graphical Analysis
5.3.1.
Cobweb Plots
5.3.1.1.
Construction Method
5.3.1.2.
Interpretation of Dynamics
5.3.1.3.
Stability Visualization
5.3.2.
Staircase Diagrams
5.4.
Periodic Orbits
5.4.1.
Period-2 Cycles
5.4.1.1.
Existence Conditions
5.4.1.2.
Stability Analysis
5.4.2.
Higher-Period Cycles
5.4.2.1.
Period-Doubling
5.4.2.2.
Stability Criteria
5.5.
Chaotic Dynamics
5.5.1.
Sensitive Dependence
5.5.2.
Aperiodic Orbits
5.5.3.
The Logistic Map
5.5.3.1.
Parameter Regimes
5.5.3.2.
Bifurcation Structure
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6. Two-Dimensional Linear Systems