Abstract
It is a readily observable fact that many physical and mathematical systems possess a degree of symmetry and that a study of this symmetry may give us valuable insight into their behaviour. It is particularly interesting that symmetric systems exist which possess non-symmetric solutions and where this solution branch arises from a symmetry breaking bifurcation on a branch of symmetric solutions. In this paper we shall study an example of such a system. Namely we shall study the symmetry breaking bifurcations (henceforth denoted as SBB’s) which occur on the radially symmetric solution branches of the following semilinear elliptic equation.
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References
C.J. Budd, A semilinear elliptic problem, Oxford, Thesis, 1986.
C.J. Budd, Semilinear elliptic equations with near critical growth rates, Proc. Roy. Soc. Edinburgh 107A (1987) 249–270.
C.J. Budd, Applications of Shilnikov theory to semilinear elliptic equations, SIAM J. Numer. Anal., to appear.
C.J. Budd, Applications of Melnikov theory to semilinear elliptic equations, 1988, in preparation.
C.J. Budd and J. Norbury, Symmetry breaking in semilinear elliptic equations with near critical growth rates, Proc. 1986 Berkeley Symposium in Elliptic Equations, 1987.
D. Chillingworth, Notes on some recent methods in bifurcation theory, Banach Centre Publ. 15 (1985) 161–174.
M. Crandall and M. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971) 321–340.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 1983).
M. Golubitsky and D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. 1 (Springer, Berlin, 1985).
J. Guckenheimer and P. Holmes, Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields (Springer, Berlin, 1983).
E. Ihrig and M. Golubitsky, Pattern selection with O (3) symmetry, Phys. 13D (1984) 1–33.
H.B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, in: P.H. Rabinowitz, Ed., Applications of Bifurcation Theory (Academic Press, New York, 1977) 359–384.
D. Sattinger, Bifurcation and symmetry breaking in applied mathematics, Bull. Amer. Math. Soc. 3 (1980) 779–819.
J. Smoller and A. Wasserman, Symmetry breaking from positive solutions of semilinear elliptic equations, Proc. 1984 Dundee Conference on Differential Equations (Springer, Berlin, 1984).
A. Vanderbauwhede, Local Bifurcation and Symmetry (Pitman, London, 1982).
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© 1990 Springer Basel AG
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Budd, C. (1990). Symmetry breaking and semilinear elliptic equations. In: Mittelmann, H.D., Roose, D. (eds) Continuation Techniques and Bifurcation Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 92. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5681-2_6
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DOI: https://doi.org/10.1007/978-3-0348-5681-2_6
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-7643-2397-4
Online ISBN: 978-3-0348-5681-2
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