1. Physics 313: Lecture 14 Wednesday, 10/08/08

2. Announcements ● No class this Monday, Oct 13, since fall break. Next class will be Wed, October 15. ● Look at details of the course project available at the URL http://www.phy.duke.edu/~hsg/313/homeworks/ http://www.phy.duke.edu/~hsg/313/homeworks/ ● Short assignment will be due next Wednesday (available Thursday). ● Finish reading Chapter 5, start reading Chapter 6.

3. Do Defects Act Like Independent Particles? Phys. Rev. Lett. 80, 2306 - 2309 (1998) “Size-Dependent Transition to High-Dimensional Chaotic Dynamics in a Two-Dimensional Excitable Medium” http://crossgroup.caltech.edu/STChaos/Bar.html http://video.google.com/videoplay?docid=4775976277056099968&vt=lf&hl=en

4. Chapter 5: Model Equations ● Models based on symmetries. ● Models based on specific physical mechanisms ● Models discrete in space or time but with continuous variables (lattices of odes, coupled map lattices) ● Models derived by empirical data analysis such as principal components (POD, SVD). ● Models with discrete variables: cellular automata.

5. The Swift-Hohenberg Ecology of Models ● Heuristic derivation, boundary conditions. Challenge for class: list all possible cubic terms... ● Properties: potential versus non-potential dynamics. ● How to relate to experiments ● Generalized models: no inversion symmetry, non-potential, mean flow, achiral dynamics

6. The Swift-Hohenberg Equation has a Lyapunov Functional ● Derivation and discussion at the blackboard. ● Implications: only non-transient states are fixed points, no periodic, quasiperiodic, or chaotic dynamics. ● Lyapunov functional acts like an energy that one can use to order relative stability of patterns. ● Functional greatly aids the analysis of topological defects like dislocations.