### MyPhysicsLab – Reaction Diffusion

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There are unusual chemical reactions that exhibit oscillating behavior that can persist for a long time. Here are simulations of such a reaction-diffusion system.
We have a two-dimensional space (eg. a thin fluid) with two chemical species U,V which react with each other, and also diffuse (spread out) on their own. The system is defined by the equations

U_{t} = λ(A)U - ω(A)V + D ∇^{2}U

V_{t} = ω(A)U - λ(A)V + D ∇^{2}V

where A^{2}=U^{2}+V^{2} and
∇^{2}=∂_{x}^{2}+∂_{y}^{2}. The λ and ω functions are defined by

λ(A) = ε - a A^{2}

ω(A) = c - β A^{2}

The movies show the concentration of only one of the two chemical species, U. The concentration is represented by colors, with blue being low concentration, through light-blue, green, yellow to red which is the highest concentration. The first four movies show various initial conditions. The parameters used are: ε=1, a=1, c=0, β=1, D=0.1.

The movies are in the Microsoft .avi format, and so are likely only playable on Windows machines.

One-armed spiral waves.

Two-armed spiral waves.

Square grid oscillating.

Parallel rows moving.

The next several movies show the effect of modifying the parameters in the defining equations of the system. The parameters used are: ε=1, a=1, c=0, β=1, D=0.1 unless otherwise noted. The initial condition is the one-armed spiral in each movie.

High diffusion, D=0.4

Clockwise rotation from setting c=1.5

No rotation from setting c=0.9

Chaos results from setting beta=2.5

Chaos results from setting a=0.25