We study a vector-valued reaction-diffusion equation with Neumann boundary conditions (u: [0,π] → ℝ2). Unlike what is observed for scalar equations, where no heteroclinic connections involving periodic solutions occur, we find that steady-state/Hopf and Hopf/Hopf mode interactions produce heteroclinic solutions connecting at least one solution of standing wave type. This is achieved by restricting a problem with periodic boundary conditions and equivariant under O(2) symmetry to a suitable fixed-point space.
For completeness, we include a description of the solutions for Hopf bifurcation and mode interactions involving Hopf bifurcation, namely, steady-state/Hopf and Hopf/Hopf.
Cite this article
Sofia B. S. D. Castro, Symmetry and bifurcation of periodic solutions in Neumann boundary value problems. Port. Math. 65 (2008), no. 3, pp. 373–385DOI 10.4171/PM/1818