Portugaliae Mathematica


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Volume 65, Issue 3, 2008, pp. 373–385
DOI: 10.4171/PM/1818

Published online: 2008-09-30

Symmetry and bifurcation of periodic solutions in Neumann boundary value problems

Sofia B. S. D. Castro[1]

(1) Universidade do Porto, Portugal

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.

Keywords: Bifurcation, limit cycles, heteroclinic connections

Castro Sofia B. S.: Symmetry and bifurcation of periodic solutions in Neumann boundary value problems. Port. Math. 65 (2008), 373-385. doi: 10.4171/PM/1818