The Dynamics of Rotating Waves in Scalar Reaction Diffusion Equations

The maximal compact attractor for the RDE <!-- MATH ${u_t} = {u_{xx}} + f(u,\,{u_x})$ --> with periodic boundary conditions is studied. It is shown that any -limit set contains a rotating wave, i.e., a solution of the form . A number of heteroclinic orbits from one rotating wave to another are constructed. Our main tool is the Nickel-Matano-Henry zero number. The heteroclinic orbits are obtained via a shooting argument, which relies on a generalized Borsuk-Ulam theorem.