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Physics 313: Lecture 9 Monday, 9/22/08
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Comments You should be reading Chapter 4.
Read Appendix 1 on elementary bifurcations, review in Strogatz if necessary. Look over Section 12.4 on using iterative and Newton's methods to find stationary nonlinear solutions to a known evolution equation. Can hand in Assignment 3 on Wed, 9/23/08. Today's topics: Quantitative comparisons of theory with experiment for reaction-diffusion systems. Stability balloons
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True or False A saturated nonlinear state is a time-independent state.
Hysteresis can never be observed for a supercritical bifurcation of a uniform state. If the evolution equation of a system is translationally invariant, the solutions of the equation are also translationally invariant. As you vary an initial state, the dynamics suddenly changes from a fixed point to a limit cycle. This is an example of a bifurcation. If a system is rotationally symmetric (isotropic) at every point, then it is translationally invariant.
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Realistic Chemical Systems
“Transition to Chemical Turbulence”, Q. Ouyang and Harry L.Swinney, Chaos 1(4): (1991).
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Amplitude of Chemical Pattern Near Onset
“Transition to Chemical Turbulence”, Q. Ouyang and Harry L.Swinney, Chaos 1(4): (1991).
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Onset of Oscillatory Dynamics
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CDIMA Reaction: Chlorine Dioxide-Iodine-Malonic Acid
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CDIMA Reaction Rates Obtained From Expt
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CDIMA Reaction-Diffusion Evolution Eqs
13 parameters or 5 dimensionless parameters a big space to explore!
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CDIMA Boundary Conditions on Fields
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Boussinesq Equations for Convection
Assumptions: velocities all small compared to speed of sound, fluid depth large compared to mean free path, temperature variation of parameters small.
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Quantitative Calculation of Uniform Fixed Point for CDIMA Reaction
Ignoring confined direction may not be a good approximation, pattern formation here is likely 3D!
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Chapter 4: What Do Linear Perturbations Grow Into?
Nonlinear saturation Stability balloons: when are periodic nonlinear stationary states linearly stable? Two-dimensional lattice states: what is possible, how to understand them. Non-ideal states: the role of defects that locally disrupt periodicity.
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Stability Balloons for Convection
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The Busse Balloon
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Testing the Busse Balloon: Cross-Roll Instability
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Testing the Busse Balloon: Zigzag Instability
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Stability Balloons for Each Lattice
In principle, one needs to calculate a stability balloon for each class of lattices: stripe, rectangular, hexagon, quasicrystal, etc. Experiments often suggest which class of lattices need to be considered and theorists often study just a few of the possible lattices, say stripes, hexagons, and rectangles.
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