The complexity of nonlinear systems often requires numerical methods for a quantitative investigation. However, to get insight into such systems analytical methods are invaluable. By considering well-defined limiting cases they allow the derivation of reduced equations that capture the essential features of the system. The reduced equations provide quantitative results in the corresponding parameter regimes, which encompass in particular transitions between qualitatively different behaviors of the system. In this class the central concepts and methods are introduced.
Bifurcation theory, center manifold theorem, separation of time scales, symmetries, pattern selection, amplitude equations, Ginzburg-Landau equations, long-wave equations, phase dynamics.
Applications to fluid flow, chemical systems, biologically motivated systems.
This class will be somewhat more advanced than 438 Interdisciplinary Nonlinear Dynamics and not quite as extensive as the previous 2-quarter sequence 412-1,2. Notes and syllabus for these classes are available at www.esam.northwestern.edu/riecke under Overview of Classes
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