Dynamical Systems Modeling and Analysis

  1. Two-Dimensional Nonlinear Systems
    1. General Framework
      1. System Form
        1. Vector Field Representation
          1. Geometric Interpretation
          2. Fixed Points
            1. Finding Equilibria
              1. Isolated Fixed Points
                1. Continuous Sets of Equilibria
                2. Linearization Analysis
                  1. Jacobian Matrix
                    1. Calculation Methods
                      1. Evaluation at Fixed Points
                      2. Local Classification
                        1. Eigenvalue Analysis
                          1. Stability Determination
                          2. Limitations of Linearization
                          3. Phase Plane Analysis
                            1. Nullclines
                              1. x-Nullclines
                                1. y-Nullclines
                                  1. Intersection Analysis
                                  2. Direction Fields
                                    1. Vector Field Plotting
                                      1. Flow Patterns
                                      2. Phase Portrait Construction
                                        1. Combining Nullclines and Direction Fields
                                          1. Trajectory Sketching
                                        2. Limit Cycles
                                          1. Definition and Properties
                                            1. Existence Conditions
                                              1. Stability Classification
                                                1. Stable Limit Cycles
                                                  1. Unstable Limit Cycles
                                                    1. Semi-stable Limit Cycles
                                                  2. Theoretical Tools
                                                    1. Poincaré-Bendixson Theorem
                                                      1. Statement and Conditions
                                                      2. Ruling Out Closed Orbits
                                                        1. Gradient Systems
                                                          1. Lyapunov Functions
                                                            1. Dulac's Criterion
                                                            2. Index Theory
                                                              1. Fixed Point Index
                                                                1. Applications to Limit Cycles

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                                                            6. Two-Dimensional Linear Systems

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                                                            8. Bifurcation Theory