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Bifurcations for Turing instability without \(\mathrm{SO}(2)\) symmetry. (English) Zbl 1136.37042
Summary: We consider the Swift-Hohenberg equation with perturbed boundary conditions. We do not a priori know the eigenfunctions for the linearized problem since the \(\mathrm{SO}(2)\) symmetry of the problem is broken by perturbation. We show that how the neutral stability curves change and, as a result, how the bifurcation diagrams change by the perturbation of the boundary conditions.
MSC:
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
37L20 Symmetries of infinite-dimensional dissipative dynamical systems
35B32 Bifurcations in context of PDEs
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