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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.
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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[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
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