Model Emergent Dynamics in Complex Systems

A book that explores the derivation of multiscale models for dynamical systems, partnering algebra and geometry with a visual approach.

A.j. Roberts (Author)

9781611973556

Paperback / softback, published 30 December 2014

760 pages
25.8 x 18 x 3.8 cm, 1.354 kg

Arising from the growing interest in dynamical systems theory, this book explores the derivation of relatively simple dynamical equations that model complex physical interactions. It unifies into one coherent approach many varied extant methods for model reduction and approximation. Using mathematical models at various levels of resolution and complexity, the book establishes the relationships between multiscale models and paradoxes, and addresses model reduction for systems, resolves initial conditions, and illuminates control. The methodology focusses on the geometric picture of both coordinate transforms and invariant manifolds in dynamical systems leading to a broad range of geometric interpretations of the modeling process that it produces. This facilitates an enhanced view of other modeling methodologies such as averaging, homogenization, multiple scales, singular perturbations, and WKB theory. Researchers at the graduate level and above who need to understand systems and modeling at different levels of resolution and complexity will find this book useful.

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.

Subject Areas: Miscellaneous items [WZ]