Model Emergent Dynamics in Complex Systems

Arising out of the growing interest in and applications of modern dynamical systems theory, this book explores how to derive relatively simple dynamical equations that model complex physical interactions. The author s objectives are to use sound theory to explore algebraic techniques, develop interesting applications, and discover general modeling principles.

Model Emergent Dynamics in Complex Systems unifies into one powerful and coherent approach the many varied extant methods for mathematical model reduction and approximation. Using mathematical models at various levels of resolution and complexity, the book establishes the relationships between such multiscale models and clarifying difficulties and apparent paradoxes and addresses model reduction for systems, resolves initial conditions, and illuminates control and uncertainty.

The basis for the author s methodology is the theory and the geometric picture of both coordinate transforms and invariant manifolds in dynamical systems; in particular, center and slow manifolds are heavily used. The wonderful aspect of this approach is the range of geometric interpretations of the modeling process that it produces - simple geometric pictures inspire sound methods of analysis and construction. Further, pictures drawn of state spaces also provide a route to better assess a model s limitations and strengths. Geometry and algebra form a powerful partnership and coordinate transforms and manifolds provide a powerfully enhanced and unified view of a swathe of other complex system modeling methodologies such as averaging, homogenization, multiple scales, singular perturbations, two timing, and WKB theory.

Audience: Advanced undergraduate and graduate students, engineers, scientists, and other researchers who need to understand systems and modeling at different levels of resolution and complexity will all find this book useful.

Contents: Part I: Asymptotic methods solve algebraic and differential equations; Chapter 1: Perturbed algebraic equations solved iteratively; Chapter 2: Power series solve ordinary differential equations; Chapter 3: A normal form of oscillations illuminate their character; Part I Summary; Part II: Center manifolds underpin accurate modeling; Chapter 4: The center manifold emerges; Chapter 5: Construct slow center manifolds iteratively; Part II Summary; Part III: Macroscale spatial variations emerge from microscale dynamics; Chapter 6: Conservation underlies mathematical modeling of fluids; Chapter 7: Cross-stream mixing causes longitudinal dispersion along pipes; Chapter 8: Thin fluid films evolve slowly over space and time; Chapter 9: Resolve inertia in thicker faster fluid films; Part III Summary; Part IV: Normal forms illuminate many modeling issues; Chapter 10: Normal-form transformations simplify evolution; Chapter 11: Separating fast and slow dynamics proves modeling; Chapter 12: Appropriate initial conditions empower accurate forecasts; Chapter 13: Subcenter slow manifolds are useful but do not emerge; Part IV Summary; Part V: High fidelity discrete models use slow manifolds; Chapter 14: Introduce holistic discretization on just two elements; Chapter 15: Holistic discretization in one space dimension; Part V Summary; Part VI: Hopf bifurcation: Oscillations within the center manifold; Chapter 16: Directly model oscillations in Cartesian-like variables; Chapter 17: Model the modulation of oscillations; Part VI Summary; Part VII: Avoid memory in modeling nonautonomous systems, including stochastic; Chapter 18: Averaging is often a good first modeling approximation; Chapter 19: Coordinate transforms separate slow from fast in nonautonomous dynamics; Chapter 20: Introducing basic stochastic calculus; Chapter 21: Strong and weak models of stochastic dynamics; Part VII Summary