1. Physics 313: Lecture 9 Monday, 9/22/08

2. 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

3. 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.

4. Realistic Chemical Systems
“Transition to Chemical Turbulence”,
Q. Ouyang and Harry L.Swinney, Chaos 1(4): (1991).

5. Amplitude of Chemical Pattern Near Onset
“Transition to Chemical Turbulence”,
Q. Ouyang and Harry L.Swinney, Chaos 1(4): (1991).

6. Onset of Oscillatory Dynamics

7. CDIMA Reaction: Chlorine Dioxide-Iodine-Malonic Acid

8. CDIMA Reaction Rates Obtained From Expt

9. CDIMA Reaction-Diffusion Evolution Eqs
13 parameters or 5 dimensionless parameters
a big space to explore!

10. CDIMA Boundary Conditions on Fields

11. 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.

12. Quantitative Calculation of Uniform Fixed Point for CDIMA Reaction
Ignoring confined direction may not be a good approximation, pattern formation here is likely 3D!

13. 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.

14. Stability Balloons for Convection

15. The Busse Balloon

16. Testing the Busse Balloon: Cross-Roll Instability

17. Testing the Busse Balloon: Zigzag Instability

18. 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.