Abstract
We investigate Hopf bifurcation of an example reaction-diffusion system on a square domain with Robin boundary conditions; the Brusselator equations. By performing a smooth homotopy of boundary conditions from Neumann to Dirichlet type, we observe the creation of branches of periodic solutions with submaximal symmetry in codimension two bifurcations, although we do not fully calculate the branching behaviour. We also note that mode interactions behave generically on varying the boundary conditions. The investigation is performed using a numerical Liapunov-Schmidt reduction technique of Ashwin, Böhmer, and Mei (1994) and an analysis of Swift (1988).
Similar content being viewed by others
References
Ashwin, P. (1992). High corank mode interactions on the square. In Allgower, E., Böhmer, K., and Golubitsky, M. (eds.),Bifurcation and Symmetry, Marburg 1991, ISNM 104, BirkhÄuser, Basel, pp. 23–33.
Ashwin, P., Böhmer, K., and Mei, Z. (1993). A numerical Liapunov-Schmidt method for finitely determined problems. In Allgower, E., Georg, K., and Miranda, R. (eds.),Exploiting Symmetry in Applied and Numerical Analysis, AMS Lect. Notes Appl. Math. Vol. 29, pp. 49–70.
Ashwin, P., Böhmer, K., and Mei, Z. (1994). A numerical Liapunov-Schmidt method with applications to Hopf bifurcation on a square.Math. Comp. (in press).
Crawford, J. D. (1991). Normal forms for driven surface waves.Physica D 52, 429–457.
Crawford, J. D., Gollub, J. P., and Lane, D. (1993). Hidden symmetries of parametrically forced waves.Nonlinearity 6, 119–164.
Dillon, R., Maini, P. K., and Othmer, H. G. (1994). Pattern formation in generalized Turing systems.J. Math. Biol. 32, 345–393.
Gardner, R. A. (1986a). Global continuation of branches of nondegenerate solutions.J. Diff. Eqs. 61, 321–334.
Gardner, R. A. (1986b). Existence of multidimensional travelling wave solutions of an initial-boundary value problem.J. Diff. Eqs. 61, 335–379.
Kuttler, J. R., and Sigillito, V. G. (1984). Eigenvalues of the Laplacian in two dimensions.SIAM Rev. 26, 163–193.
Mei, Z. (1990). Bifurcations of a simplified buckling problem and the effect of discretizations.Manuscripta Math. 71, 225–252.
Prigogine, I., and Glansdorff, P. (1971).Structure, Stabilité et Fluctuations, Masson, Paris.
Rabier, P. J. (1985).Topics in One-Parameter Bifurcation Problems, Springer-Verlag, Heidelberg.
Shaw, G. B. (1974). Degeneracy in the particle-in-a-box problem.J. Phys. A Math. Nucl. Gen. 7, 1537–1546.
Swift, J. W. (1988). Hopf bifurcation with the symmetry of the square.Nonlinearity 1, 333–377.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ashwin, P., Zhen, M. A Hopf bifurcation with Robin boundary conditions. J Dyn Diff Equat 6, 487–505 (1994). https://doi.org/10.1007/BF02218859
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02218859
Key words
- Liapunov-Schmidt method
- Robin boundary condition
- Hopf bifurcation with square symmetry
- spatiotemporal pattern formation