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

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.

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?

Granular Crispation Experiments: Why Dynamics Similar to Convection?

Chemical Reaction-Diffusion Patterns Complex Patterns in a Simple System, John Pearson, Science 261, (1993)

Snowflakes! (Dendritic Growth)

Magnification of Molecular Noise by Convective Instability

Belousov-Zhabotinsky Reaction: Targets and Spirals

Spiral Waves in Dictyostelium Discoideum

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

Dictyostelium Discovered in NC State amoeba?

Spiral Waves and Fibrillating Hearts?

Or Three-Dimensional Cardiac Dynamics?

Gentle Control Algorithms for Heart as Chaotic System? Science 257: (1992)

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

Taylor-Couette Flow

“Phase Diagram” for Taylor-Couette Flow

Domain Chaos, Spiral Defect Chaos Bifurcation Diagram

Compare with “Phase Diagram” For Logistic Map

How High-Dimensional Systems Might Arise

Casual Criteria for High-Dimensional Dynamics

Quantifying Complexity: Correlation Lengths and Fractal Dimensions

Linear Instability Analysis of Time-Independent Spatially Uniform States

Growth Rate For Swift-Hohenberg State u=0

Neutral Stability Region for Swift-Hohenberg

General Mathematical Strategy for a Linear Stability Analysis

Experimental Linear Stability Analysis: Study Weakly Nonlinear Stationary States

Empirical Estimate of Critical Temperature

Taylor-Couette: Theory Versus Experiment