Model Emergent Dynamics in Complex Systems

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.

• Part I: Asymptotic methods solve algebraic and differential equationsChapter 1: perturbed algebraic equations solved iterativelyChapter 2: power series solve ordinary differential equationsChapter 3: A normal form of oscillations illuminate their characterPart I SummaryPart II: Center manifolds underpin accurate modelingChapter 4: The center manifold emergesChapter 5: Construct slow center manifolds iterativelyPart II SummaryPart III: Macroscale spatial variations emerge from microscale dynamicsChapter 6: Conservation underlies mathematical modeling of fluidsChapter 7: Cross-stream mixing causes longitudinal dispersion along pipesChapter 8: Thin fluid films evolve slowly over space and timeChapter 9: Resolve inertia in thicker faster fluid filmsPart III SummaryPart IV: Normal forms illuminate many modeling issuesChapter 10: Normal-form transformations simplify evolutionChapter 11: Separating fast and slow dynamics proves modelingChapter 12: Appropriate initial conditions empower accurate forecastsChapter 13: Subcenter slow manifolds are useful but do not emergePart IV SummaryPart V: High fidelity discrete models use slow manifoldsChapter 14: Introduce holistic discretization on just two elementsChapter 15: Holistic discretization in one space dimensionPart V SummaryPart VI: Hopf bifurcation: Oscillations within the center manifoldChapter 16: Directly model oscillations in Cartesian-like variablesChapter 17: Model the modulation of oscillationsPart VI SummaryPart VII: Avoid memory in modeling nonautonomous systems, including stochasticChapter 18: Averaging is often a good first modeling approximationChapter 19: Coordinate transforms separate slow from fast in nonautonomous dynamicsChapter 20: Introducing basic stochastic calculusChapter 21: Strong and weak models of stochastic dynamicsPart VII Summary