Course description:
Patterns (sand ripples, stripes or spots on animal coats, ...) are
ubiquitous in nature. A striking feature of pattern-forming systems is
the similarity in the patterns they exhibit in spite of the
differences in the physical or biological nature of these systems. A
century of experimentation and a few decades of modeling/analysis have
brought some answers to this puzzling phenomenon. In particular, the
near-threshold theory of pattern formation, which involves multiple
scales analysis and which has its roots in bifurcation theory for
dynamical systems, has now become a standard tool in the analysis and
modeling of nonlinear phenomena.
This course will discuss near- and far-from-threshold pattern
formation. Focus will be placed on theoretical issues, but both experiments
in and numerical simulations of pattern-forming systems will be discussed.
The main topics are listed below.
- Pattern formation in nature and in laboratory experiments.
- Patterns near threshold: bifurcation theory, multiple scales
analysis, envelope equations.
- Applications to pattern formation in biological systems.
- Patterns far from threshold and the phase-diffusion equation.
- Order-parameter equations as partial differential equations:
analytical solutions, linear stability analysis, complex dynamics.
- (time-permitting) On the validity of order-parameter equations.
Prerequisites for this course include good undergraduate knowledge of
partial and ordinary differential equations, linear algebra, and numerical
analysis. Students will also be expected to have some knowledge of one
of these topics at the graduate level.
