Methods of Nonlinear Analysis

ESAM D-12-1,2

Fall/Winter 1999

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.

Main topics:

Bifurcation theory, center manifold theorem, separation of time scales, symmetries, pattern selection, amplitude equations, Ginzburg-Landau equations, long-wave equations, phase dynamics, front dynamics, solitary waves, secondary bifurcations. Applications to fluid flow, chemical systems, biologically motivated systems.

Prerequisites:

Ordinary differential equations, some knowledge in perturbation methods and in partial differential equations.

Time and Place:

MW 2-3:30 TCH M177