Pattern formation arising from interactions between Turing and wave instabilities

We study pattern formation arising from the interaction of the stationary Turing and wave ~oscillatory Turing! instabilities. Interaction and competition between these symmetry-breaking modes lead to the emergence of a large variety of spatiotemporal patterns, including modulated Turing structures, modulated standing waves, and combinations of Turing structures and spiral waves. Spatial resonances are obtained near codimension-two Turing-wave bifurcations. Far from bifurcation lines, we obtain inwardly propagating spiral waves with Turing spots at their tips. We demonstrate that the coexistence of Turing spots and traveling waves is a result of interaction between Turing and oscillatory modes, while the inwardly propagating waves ~antispirals! do not require this interaction; they can arise from the wave instability combined with a negative group velocity. © 2002 American Institute of Physics. @DOI: 10.1063/1.1507110#