Dynamical Systems Modeling and Analysis

  1. Two-Dimensional Linear Systems
    1. Mathematical Framework
      1. Matrix Form x' = Ax
        1. Solution Structure
          1. Fundamental Matrix
          2. Phase Plane Concepts
            1. Phase Plane Coordinates
              1. Trajectory Representation
                1. Initial Conditions
                2. Eigenvalue Analysis
                  1. Characteristic Equation
                    1. Eigenvalue Calculation
                      1. Eigenvector Determination
                      2. Classification by Eigenvalues
                        1. Real Distinct Eigenvalues
                          1. Saddle Points
                            1. Stable Nodes
                              1. Unstable Nodes
                              2. Complex Conjugate Eigenvalues
                                1. Spiral Points
                                  1. Centers
                                    1. Stability Determination
                                    2. Repeated Eigenvalues
                                      1. Star Nodes
                                        1. Degenerate Nodes
                                          1. Jordan Block Cases
                                        2. Phase Portrait Construction
                                          1. Eigenvector Directions
                                            1. Trajectory Sketching
                                              1. Stability Regions
                                              2. Special Cases
                                                1. Zero Eigenvalues
                                                  1. Pure Imaginary Eigenvalues
                                                    1. Lines of Equilibria

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                                                  5. One-Dimensional Discrete Systems

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                                                  7. Two-Dimensional Nonlinear Systems