Model Emergent Dynamics in Complex Systems
Published on 21. May 2015
Book
Paperback/Softback
760 pages
978-1-61197-355-6 (ISBN)
Description
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.
More details
Person
A. J. Roberts is a Professor and Chair in the School of Mathematical Sciences at the University of Adelaide. He has lectured and conducted research at the University of New South Wales and the University of Southern Queensland and has published over 100 refereed international journal articles. As a leader in developing and applying a branch of modern dynamical systems theory, in conjunction with new computer algebra algorithms in scientific computing, Professor Roberts derives and interprets mathematical and computational models of complex multiscale systems, both deterministic and stochastic.
Content
- 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