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Physics 313: Lecture 6 Wednesday, 9/10/08 Comments to the Class ● ● You should finish reading Chapter 3 of Cross-Greenside and start Chapter 4. Topics: – Type I, II, and III instabilities, each 's' or 'o': six categories in all. – Chapter 3: Turing-instability of two interacting chemicals. Vocabulary Check ● Extended, confined coordinates? ● Primary, secondary, tertiary instabilities? ● Reduced bifurcation parameter? ● Neutral stability curve? ● Coherence length? ● Type “s” or type “o” instability? ● Translationally or rotationally invariant system, equation, state? Type I, II, III instabilities Stationary “s” or oscillatory “o” Note: growth rate Re[q] typically has multiple extrema but typically only one extremum crosses zero first. Need to vary two parameters to get two extrema to cross at same time. Seashells As Illustration of Instability Types Neutral Stability Region Re[q] =0 For Swift-Hohenberg Linear Instability of Single-Reagent Reaction-Diffusion System ● How does analysis change if we allow two extended coordinates, c=c(x,y,t)? Growth Rate for Isotropic Systems Pattern of Random Critical States nstates = 8 angles = Table[ Random[ Real, {0, 2 Pi} ], {nstates} ] phases = Table[ Random[ Real, {0, 2 Pi} ], {nstates} ] amps = Table[ Random[ Real, {0, 1.0} ], {nstates} ] f2d[x_, y_] := Sum[ amps[[i]] Cos[ Cos[angles[[i]]] x + Sin[angles[[i]]] y + phases[[i]]] , {i, 1, nstates} ] DensityPlot[ f2d[x,y], {x,0,10Pi}, {y,0,10Pi}, PlotPoints -> 50 ] Growth Rate Surfaces For Uniaxial Anisotropic Medium Nematic liquid crystal Growth Rate for Anisotropic Uniaxial System