Model Emergent Dynamics in Complex Systems

Preface; Part I. Asymptotic Methods Solve Algebraic and Differential Equations: 1. Perturbed algebraic equations solved iteratively; 2. Power series solve ordinary differential equations; 3. A normal form of oscillations illuminate their character; Part I summary; Part II. Center Manifolds Underpin Accurate Modeling: 4. The center manifold emerges; 5. Construct slow center manifolds iteratively; Part II summary; Part III. Macroscale Spatial Variations Emerge from Microscale Dynamics: 6. Conservation underlies mathematical modeling of fluids; 7. Cross-stream mixing causes longitudinal dispersion along pipes; 8. Thin fluid films evolve slowly over space and time; 9. Resolve inertia in thicker faster fluid films; Part III summary; Part IV. Normal Forms Illuminate Many Modeling Issues: 10. Normal-form transformations simplify evolution; 11. Separating fast and slow dynamics proves modeling; 12. Appropriate initial conditions empower accurate forecasts; 13. Subcenter slow manifolds are useful but do not emerge; Part IV summary; Part V. High Fidelity Discrete Models Use Slow Manifolds: 14. Introduce holistic discretization on just two elements; 15. Holistic discretization in one space dimension; Part V summary; Part VI. Hopf Bifurcation: Oscillations Within the Center Manifold: 16. Directly model oscillations in Cartesian-like variables; 17. Model the modulation of oscillations; Part VI summary; Part VII. Avoid Memory in Modeling Nonautonomous Systems, Including Stochastic: 18. Averaging is often a good first modeling approximation; 19. Coordinate transforms separate slow from fast in nonautonomous dynamics; 20. Introducing basic stochastic calculus; 21. Strong and weak models of stochastic dynamics; Part VII summary; Bibliography; Index.