Nonlinear Dynamics and Chaos

  1. Two-Dimensional Flows
    1. Phase Plane Analysis
      1. Construction of Phase Portraits
        1. Nullclines and Isoclines
          1. Direction Fields
            1. Trajectory Behavior
            2. Linear Systems in Two Dimensions
              1. Matrix Representation
                1. Eigenvalue and Eigenvector Analysis
                  1. Classification of Linear Fixed Points
                    1. Saddle Points
                      1. Nodes
                        1. Stable Nodes
                          1. Unstable Nodes
                            1. Degenerate Nodes
                            2. Spiral Points
                              1. Stable Spirals (Foci)
                                1. Unstable Spirals (Foci)
                                2. Centers
                                3. Trace-Determinant Classification
                                  1. Phase Portraits for Linear Systems
                                  2. Nonlinear Systems in Two Dimensions
                                    1. Local Linearization
                                      1. Jacobian Matrix
                                        1. Linear Approximation Near Fixed Points
                                        2. Hartman-Grobman Theorem
                                          1. Nonlinear Phase Portraits
                                            1. Global vs. Local Behavior
                                              1. Separatrices and Stable/Unstable Manifolds
                                              2. Periodic Orbits and Limit Cycles
                                                1. Definition of Limit Cycles
                                                  1. Stability of Periodic Orbits
                                                    1. Poincaré Maps
                                                      1. Poincaré-Bendixson Theorem
                                                        1. Statement and Proof Outline
                                                          1. Applications and Limitations
                                                          2. Criteria for Ruling Out Closed Orbits
                                                            1. Dulac's Criterion
                                                              1. Bendixson's Criterion
                                                            2. Bifurcations in Two-Dimensional Systems
                                                              1. Codimension-One Bifurcations
                                                                1. Saddle-Node Bifurcation
                                                                  1. Transcritical Bifurcation
                                                                    1. Pitchfork Bifurcation
                                                                      1. Hopf Bifurcation
                                                                        1. Supercritical Hopf
                                                                          1. Subcritical Hopf
                                                                        2. Global Bifurcations
                                                                          1. Homoclinic Bifurcations
                                                                            1. Heteroclinic Bifurcations
                                                                              1. Saddle-Node Bifurcation of Cycles
                                                                              2. Bifurcation Analysis Techniques
                                                                                1. Center Manifold Theory
                                                                                  1. Normal Form Theory

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                                                                              2. One-Dimensional Flows

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                                                                              4. Higher-Dimensional Systems