# Article

Full entry | PDF   (1.2 MB)
Keywords:
perturbed boundary conditions; imperfect pitchfork bifurcation; Turing instability
Summary:
In this paper, we consider the Swift–Hohenberg equation with perturbed boundary conditions. We do not a priori know the eigenfunctions for the linearized problem since the ${\rm SO(2)}$ 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.
References:
[1] Carr J.: Applications of Center Manifold Theory. Springer–Verlag, Berlin 1981 MR 0635782
[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 MR 1279745 | Zbl 0829.92001
[3] Kabeya Y., Morishita, H., Ninomiya H.: Imperfect bifurcations arising from elliptic boundary value problems. Nonlinear Anal. 48 (2002), 663–684 DOI 10.1016/S0362-546X(00)00205-4 | MR 1868109 | Zbl 1017.34041
[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 DOI 10.1143/JPSJ.71.677 | Zbl 1161.76483
[6] Nishiura Y.: Far-from-Equilibrium Dynamics, Translations of Mathematical Monographs 209, Americal Mathematical Society, Rhode Island 200. MR 1903642
[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 DOI 10.1016/0167-2789(90)90113-4 | MR 1078607 | Zbl 0721.35008

Partner of