Model Emergent Dynamics in Complex Systems

Model Emergent Dynamics in Complex Systems unifies into one 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; clarifies difficulties and apparent paradoxes and addresses model reduction for systems; resolves initial conditions, and illuminates control and uncertainty.

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

The methodology throughout the book is the theory and geometric picture of both coordinate transforms and invariant manifolds in dynamical systems; in particular, center and slow manifolds are heavily used. The benefit to 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 are combined to coordinate transforms, and manifolds provide an enhanced and unified view of other complex system modeling methodologies such as: averaging, homogenization, multiple scales, singular perturbations, two timing, and WKB theory.

A set of MATLAB code examples is available throughout the book.