1. Physics 313: Lecture 3 Monday, 9/1/08

2. Comments to the Class ● All students, enrolled and auditing: please do Problem 1 of Assignment 1. ● You should be reading Chapter 2 of Cross-Greenside (linear stability). ● Typos, errors, corrections in book? Please forward... ● Topics: – Finish discussion of examples: what are the interesting scientific questions? – When are natural systems high-dimensional and so need new approaches? – Linear stability of spatially uniform fixed points: a first step toward understanding spatially extended driven dissipative systems.

3. Discussion Questions ● From reading Chapter 1: what is a key scientific reason why spatially extended nonequilibrium systems are hard to understand? (A mathematical reason is the presence of nonlinearities.) ● How is it possible to talk about a time-dependent spatially- varying temperature field T(t,x,y,z) (e.g., in the heat diffusion equation ∂ t T = · ∇ 2 T ) when T is a property of systems in thermodynamic equilibrium for which T is constant in time and space? ● Estimate order of magnitude of fluid speed in convecting flow?

4. Granular Crispation Experiments: Why Dynamics Similar to Convection?

5. Chemical Reaction-Diffusion Patterns Complex Patterns in a Simple System, John Pearson, Science 261, 189-192 (1993) http://www.youtube.com/watch?v=BfcQI7NhZqw

6. Snowflakes! (Dendritic Growth) http://www.its.caltech.edu/~atomic/snowcrystals/photos/photos.htm

7. Magnification of Molecular Noise by Convective Instability

8. Belousov-Zhabotinsky Reaction: Targets and Spirals http://www.youtube.com/watch?v=bH6bRt4XJcw http://www.youtube.com/watch?v=GEF_NtTNeMc&feature=related

9. Spiral Waves in Dictyostelium Discoideum

10. Dicty Important for Understanding Transition from Single- to Multi-cell Organisms

11. Dictyostelium Discovered in NC State amoeba?

12. Spiral Waves and Fibrillating Hearts?

13. Or Three-Dimensional Cardiac Dynamics? http://thevirtualheart.org/

14. Gentle Control Algorithms for Heart as Chaotic System? Science 257:1230-1235 (1992)

15. Anti-Chaos Control for Epilepsy? Are Brain Dynamics Chaotic, Controllable?

16. Taylor-Couette Flow

17. “Phase Diagram” for Taylor-Couette Flow

18. Domain Chaos, Spiral Defect Chaos Bifurcation Diagram

19. Compare with “Phase Diagram” For Logistic Map

20. How High-Dimensional Systems Might Arise http://www.cmp.caltech.edu/~mcc/Chaos_Course/Demonstrations.html http://www.geom.uiuc.edu/~worfolk/apps/Lorenz/

21. Casual Criteria for High-Dimensional Dynamics

22. Quantifying Complexity: Correlation Lengths and Fractal Dimensions

23. Linear Instability Analysis of Time-Independent Spatially Uniform States

24. Growth Rate For Swift-Hohenberg State u=0

25. Neutral Stability Region for Swift-Hohenberg

26. General Mathematical Strategy for a Linear Stability Analysis

27. Experimental Linear Stability Analysis: Study Weakly Nonlinear Stationary States

28. Empirical Estimate of Critical Temperature

29. Taylor-Couette: Theory Versus Experiment