Physics 313: Lecture 9
Monday, 9/22/08
Comments
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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.
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Can hand in Assignment 3 on Wed, 9/23/08.
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Today's topics:
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Quantitative comparisons of theory with experiment
for reaction-diffusion systems.
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Stability balloons
True or False
1) A saturated nonlinear state is a time-independent
state.
2) Hysteresis can never be observed for a supercritical
bifurcation of a uniform state.
3) If the evolution equation of a system is
translationally invariant, the solutions of the equation
are also translationally invariant.
4) 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.
5) If a system is rotationally symmetric (isotropic) at
every point, then it is translationally invariant.
Realistic Chemical Systems
“Transition to Chemical Turbulence”,
Q. Ouyang and Harry L.Swinney, Chaos 1(4):411-420 (1991).
Amplitude of Chemical Pattern Near Onset
“Transition to Chemical Turbulence”,
Q. Ouyang and Harry L.Swinney, Chaos 1(4):411-420 (1991).
Onset of Oscillatory Dynamics
CDIMA Reaction:
Chlorine Dioxide-Iodine-Malonic Acid
CDIMA Reaction Rates Obtained From Expt
CDIMA Reaction-Diffusion Evolution Eqs
13 parameters or 5 dimensionless parameters
a big space to explore!
CDIMA Boundary Conditions on Fields
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.
Quantitative Calculation of Uniform Fixed
Point for CDIMA Reaction
Ignoring confined direction
may not be a good
approximation, pattern
formation here is likely 3D!
Chapter 4: What Do Linear Perturbations
Grow Into?
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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.
Stability Balloons for Convection
The Busse Balloon
Testing the Busse Balloon:
Cross-Roll Instability
Testing the Busse Balloon:
Zigzag Instability
Stability Balloons for Each Lattice
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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.