Rotating Chemical Waves in the Gray–Scott Model

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 $O(2)$ symmetry is investigated. Using a hybrid numerical/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.