Ogawa, Toshiyuki; Okuda, Takashi Bifurcations for Turing instability without \(\mathrm{SO}(2)\) symmetry. (English) Zbl 1136.37042 Kybernetika 43, No. 6, 869-877 (2007). 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 Keywords:Swift-Hohenberg equation with perturbed boundary conditions; neutral stability curves; bifurcation diagrams; imperfect pitchfork bifurcation; linearized eigenvalue problem PDF BibTeX XML Cite \textit{T. Ogawa} and \textit{T. Okuda}, Kybernetika 43, No. 6, 869--877 (2007; Zbl 1136.37042) Full Text: Link EuDML References: [1] Carr J.: Applications of Center Manifold Theory. Springer-Verlag, Berlin 1981 · Zbl 0464.58001 [2] Dillon R., Maini P. K., Othmer H. G.: Pattern formation in generalized Turing systems I. Steady-state patterns in systems with mixed boundary conditions. J. Math. Biol. 32 (1994), 345-393 · Zbl 0829.92001 · doi:10.1007/BF00160165 [3] Kabeya Y., Morishita, H., Ninomiya H.: Imperfect bifurcations arising from elliptic boundary value problems. Nonlinear Anal. 48 (2002), 663-684 · Zbl 1017.34041 · doi:10.1016/S0362-546X(00)00205-4 [4] Kato Y., Fujimura K.: Folded solution branches in Rayleigh-Bénard convection in the presence of avoided crossings of neutral stability curves. J. Phys. Soc. Japan 75 (2006), 3, 034401-034405 · doi:10.1143/JPSJ.75.034401 [5] Mizushima J., Nakamura T.: Repulsion of eigenvalues in the Rayleigh-Bénard problem. J. Phys. Soc. Japan 71 (2002), 3, 677-680 · Zbl 1161.76483 · doi:10.1143/JPSJ.71.677 [6] Nishiura Y.: Far-from-Equilibrium Dynamics, Translations of Mathematical Monographs 209, Americal Mathematical Society, Rhode Island 200. · Zbl 1013.37001 [7] Ogawa T., Okuda T.: Bifurcation analysis to Swift-Hohenberg equation with perturbed boundary conditions. In preparation · Zbl 1221.37157 [8] Tuckerman L., Barkley D.: Bifurcation analysis of the Eckhaus instability. Phys. D 46 (1990), 57-86 · Zbl 0721.35008 · doi:10.1016/0167-2789(90)90113-4 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.