Spatiotemporal Dynamics and Pattern
Formation
with Examples from Physics,
Chemistry, Biology, and Engineering
Instructor
Roman Grigoriev
Office: Howey W304 (office hours: TBA)
Phone: (404) 385-1130
E-mail:
Place and Times
Course Description
Most macroscopic structures or patterns we see around us (clouds, ocean waves, sand dunes, zebra stripes, contraction waves of the heart muscle) arise as a result of competition between external driving and internal dissipation. Although the nature of pattern forming systems can be very different, essentially all of them possess a number of universal traits, which allows a unified treatment of their dynamics. This course will develop a general theory of pattern formation in physical, chemical, and biological systems using stability theory, perturbation theory, and symmetry analysis.
This is a graduate level course intended for math, science and engineering
students. Good knowledge of partial and ordinary differential equations and
linear algebra is the main prerequisite.
Textbook
The main textbook is
- Michael Cross and Henry Greenside, Pattern Formation and Dynamics in
Nonequilibrium Systems (Cambridge University Press, 2009)
There are a few other good books that have a somewhat narrower focus:
- Paul Manneville, Dissipative structures and weak turbulence
(Academic Press, 1990)
- Daniel Walgraef, Spatio-Temporal Pattern Formation: With Examples from
Physics, Chemistry, and Materials Science (Springer-Verlag, 1997).
- James Murray, Mathematical Biology, Vol. 2 (Springer-Verlag, 2002).
There is also a collection of review articles on this subject:
- Cross and Hohenberg, Pattern formation outside of equilibrium,
Reviews of Modern Physics 65, 851 (1993).
- Koch and Meinhardt, Biological pattern formation: from basic mechanisms
to complex structures, Reviews of Modern Physics 66, 1481 (1994).
- Merzhanov and Rumanov, Physics of reaction waves, Reviews of Modern
Physics 71,
1173 (1999).
- Aranson and Kramer, The world of the complex Ginzburg-Landau equation,
Reviews of Modern Physics 74, 99 (2002).
Grading
There will be two exams (a mid-term and a final). The grades will be based on the
homeworks (33%) and the exams (midterm 33%, final 34%). Homework assignments will be posted on the web every Friday and will be due next Friday in class. You are welcome to discuss problems with each other, but the solutions have to be executed and submitted individually. In general you are expected to comply with the academic honor code.
Course Outline
1. Patterns in nature and in the lab
2. Linear stability analysis
3. Applications of linear stability analysis
4. Patterns and symmetry
5. Secondary instabilities
6. Amplitude equations
7. Traveling waves and fronts
Homework Assignments