There are unusual chemical reactions that exhibit oscillating behavior that can persist for a long time. Below are links to videos of simulations of such a reaction-diffusion system. The simulations were done using Matlab, see the paper for details.

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 diffusion constant is
*D*
. 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 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

This web page was first published February 2002.