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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
7.
Two-Dimensional Nonlinear Systems
7.1.
General Framework
7.1.1.
System Form
7.1.2.
Vector Field Representation
7.1.3.
Geometric Interpretation
7.2.
Fixed Points
7.2.1.
Finding Equilibria
7.2.2.
Isolated Fixed Points
7.2.3.
Continuous Sets of Equilibria
7.3.
Linearization Analysis
7.3.1.
Jacobian Matrix
7.3.1.1.
Calculation Methods
7.3.1.2.
Evaluation at Fixed Points
7.3.2.
Local Classification
7.3.2.1.
Eigenvalue Analysis
7.3.2.2.
Stability Determination
7.3.3.
Limitations of Linearization
7.4.
Phase Plane Analysis
7.4.1.
Nullclines
7.4.1.1.
x-Nullclines
7.4.1.2.
y-Nullclines
7.4.1.3.
Intersection Analysis
7.4.2.
Direction Fields
7.4.2.1.
Vector Field Plotting
7.4.2.2.
Flow Patterns
7.4.3.
Phase Portrait Construction
7.4.3.1.
Combining Nullclines and Direction Fields
7.4.3.2.
Trajectory Sketching
7.5.
Limit Cycles
7.5.1.
Definition and Properties
7.5.2.
Existence Conditions
7.5.3.
Stability Classification
7.5.3.1.
Stable Limit Cycles
7.5.3.2.
Unstable Limit Cycles
7.5.3.3.
Semi-stable Limit Cycles
7.6.
Theoretical Tools
7.6.1.
Poincaré-Bendixson Theorem
7.6.1.1.
Statement and Conditions
7.6.2.
Ruling Out Closed Orbits
7.6.2.1.
Gradient Systems
7.6.2.2.
Lyapunov Functions
7.6.2.3.
Dulac's Criterion
7.6.3.
Index Theory
7.6.3.1.
Fixed Point Index
7.6.3.2.
Applications to Limit Cycles
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6. Two-Dimensional Linear Systems
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8. Bifurcation Theory