Dynamical Systems Modeling and Analysis

  1. Numerical Methods and Computational Analysis
    1. Numerical Integration of ODEs
      1. Euler's Method
        1. Forward Euler
          1. Backward Euler
            1. Error Analysis
            2. Runge-Kutta Methods
              1. Second-Order Methods
                1. Fourth-Order Runge-Kutta
                  1. Adaptive Step Size
                  2. Multi-Step Methods
                    1. Adams-Bashforth Methods
                      1. Adams-Moulton Methods
                      2. Stiff Equations
                        1. Implicit Methods
                          1. Stability Regions
                        2. Computational Tools for Analysis
                          1. Time Series Generation
                            1. Phase Portrait Plotting
                              1. Bifurcation Diagram Construction
                                1. Lyapunov Exponent Calculation
                                  1. Fractal Dimension Estimation
                                  2. Numerical Challenges
                                    1. Step Size Selection
                                      1. Accuracy versus Efficiency
                                        1. Error Control
                                        2. Numerical Stability
                                          1. Round-off Errors
                                            1. Truncation Errors
                                            2. Chaos Detection
                                              1. Distinguishing Chaos from Noise
                                                1. Numerical Artifacts
                                                2. Long-Time Integration
                                                  1. Shadowing Lemma
                                                    1. Computational Limits

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                                                11. Applications in Biology