Spatiotemporal Dynamics and Pattern Formation
with Examples from Physics, Chemistry, and Biology

Instructor

Roman Grigoriev
Office: Howey W304 (office hours: Wednesday 1-2pm)
Phone: (404) 385-1130
E-mail:

Monday, Wednesday & Friday, 11am-12pm
Room S106, Howey Physics Building

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 using stability theory, bifurcations and symmetry analysis using a variety of physical, chemical, and biological systems as examples.

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.

There is not a textbook that would be sutable for the whole course, but there are a few good books that cover some aspects of it:
• 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).
• Philip Ball, The Self-made Tapestry: Pattern Formation in Nature (available electonically at NetLibrary within GT)
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).

There will be a mid-term and a final exam and the grades will be based on the exams and homeworks (33%/33%/33%). Homework assignments will be posted on the web every Friday and will be due next Friday in class. You can 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.

January 9
1. Introduction
Reading: lecture notes
Related websites: Solar granular convection

January 11
2. Patterns in nature
Reading: lecture notes
Related websites: Snowflakes, Slime mold aggregation

January 13
3. Prepared patterns
Reading: lecture notes

January 16
Institutional holiday

January 18
4. Prepared patterns
Reading: lecture notes
Related websites: Spiral defect chaos

January 20
5. Nondimensionalization
Reading: lecture notes
Related websites: Granular crispation

January 23
6. Linear stability analysis
Reading: lecture notes

January 25
7. Linear stability of extended systems
Reading: lecture notes

January 27
8. Boundary conditions
Reading: lecture notes
Problems: assignment #1, solutions

January 30
9. Swift-Hohenberg equation
Reading: lecture notes

February 1
10. Classification of linear instabilities
Reading: lecture notes

February 3
11. Classification of linear instabilities
Reading: lecture notes
Problems: assignment #2, solutions

February 6
12. Types of physical boundaries
Reading: lecture notes

February 8
13. Reaction-diffusion systems
Reading: lecture notes

February 10
14. CDIMA reaction
Reading: lecture notes
Problems: assignment #3, solutions

February 13
15. Turing instability
Reading: lecture notes, Biological Pattern Formation by Koch and Meinhardt
Matlab simulation: Brusselator

February 15
16. Rayleigh-Benard convection
Reading: lecture notes

February 17
17. Rayleigh-Benard convection
Reading: lecture notes
Problems: assignment #4, solutions

February 20
18. Rayleigh-Benard convection
Reading: lecture notes

February 22
19. Rayleigh-Benard convection
Reading: lecture notes

February 24
20. D'Arcy convection
Reading: lecture notes
Problems: assignment #5, solutions

February 27
21. Qualitative features of nonlinear states
Reading: lecture notes

March 1
22. Patterns and symmetry
Reading: lecture notes

March 3
23. Bifurcation theory for extended systems
Reading: lecture notes
Problems: assignment #6, solutions

March 6
24. Nonlinear saturation in Swift-Hohenberg equation
Reading: lecture notes

March 8
25. Nonlinear saturation in d'Arcy convection
Reading: lecture notes

March 10
26. Secondary instabilities and stability baloons
Reading: lecture notes
Problems: assignment #7, solutions

Midterm (due 3/17/06)
Click here to download the assignment and solutions

March 13
27. Long-wavelength instabilities
Reading: lecture notes
Matlab simulations: zigzag, Eckhaus, cross-roll, skew-varicose instability

March 15
28. Short-wavelength instabilities
Reading: Maple code

March 17
29. Short-wavelength instabilities
Reading: Maple code
Problems: assignment #8, solutions

March 20-24
Spring Break

March 27
30. Bousse baloon
Reading: lecture notes

March 29
31. Amplitude equations for stripe states
Reading: lecture notes

March 31
32. Phenomenological derivation
Reading: lecture notes
Problems: assignment #9, solutions

April 3
33. Parameters in the amplitude equations
Reading: lecture notes

April 5
34. Universality and scales
Reading: lecture notes

April 7
35. Boundary conditions
Reading: lecture notes
Problems: assignment #10, solutions

April 10
36. Amplitude equations for lattice states
Reading: lecture notes

April 12
37. Stability baloon revisited
Reading: lecture notes

April 14
38. Moving fronts
Reading: lecture notes
Problems: assignment #11, solutions

April 17
39. Front selection
Reading: lecture notes

April 21
40. Pattern selection by moving fronts
Reading: lecture notes

April 24
41. Dynamics of excitable media
Reading: lecture notes

April 26
42. Waves in excitable media
Reading: lecture notes
Matlab simulation: Spiral waves in the FitzHugh-Nagumo model and Complex Ginzburg-Landau equation.

April 28
43. Stability of traveling waves
Reading: lecture notes

Final (due 5/4/06)
Click here to download the assignment and solutions

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