W. Farr and M. Golubitsky
Rotating chemical waves in the Gray-Scott model
SIAM J. Appl. Math. . 52 No.1 (1992) 181-221.
A set of reaction-diffusion equations is considered, known as the Gray-
Scott model, defined on a circle, and the stability of rotating wave
solutions formed via Hopf bifurcations that break the circular 0(2)
symmetry is investigated. Using a hybrid numberical/analytical technique,
center manifold/normal form reductions are performed to analyze symmetry-
breaking Hopf bifurcations, degenerate Hopf bifurcations, and Hopf-Hopf
mode interactions. It is found that stable rotating waves exist over
broad ranges of parameter values and that the bifurcation behavior of
this relatively simple model can be quite complex, e.g., two- and
three-frequency motions exist.