The dynamics of rotating waves in scalar reaction diffusion equations

Abstract

The maximal compact attractor for the RDE u t = u x x + f ( u , u x ) {u_t} = {u_{xx}} + f(u,\,{u_x}) with periodic boundary conditions is studied. It is shown that any ω \omega -limit set contains a rotating wave, i.e., a solution of the form U ( x − c t ) U(x - ct) . 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.