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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

106 V Vanderbauwhede, A.

106 V Vanderbauwhede, A. Local Bifurcations and Symmetry, Pitman Research Notes in Math. 75 (1982), Pitman Publ. I~ndon W Wahl, J. The Jacobian algebra of a graded Gorenstein singularity, Duke Jour. Math. 55 (1987) 843-871 ADDRESS Department of Mathematics University of North Carolina Chapel Hill, N.C. 27599 U.S.A.

On a Codimension-four Bifurcation Occurring in Optical Bistability G.Dangelmayr and M.Wegelin Abstract The subject of this paper is the unfolding of a singularity of vector fields in which a cusp and a degenerate Hopf bifurcation coalesce. This singularity has codimension four and appears in the mean field equations underlying optically bistable systems. We discuss the singularities of eodimension smaller than four that occur as subsidiary bi- furcations of the unfolding and present a two-dimensional section through the stability diagram. 1 Introduction A well-known phenomenon in parameter dependent systems of ordinary differential equa- tions is the Hopf bifurcation of a periodic orbit from an equilibrium [13,20]. Generically, if only one parameter is varied, the Hopf bifurcation is non-degenerate which means that a unique periodic orbit is created when the bifurcation parameter passes through its critical value. The stability of the periodic orbit depends on a certain coefficient that has to be calculated from the underlying vector field. When a second parameter is varied this coeffi- cient may vanish and we encounter a degenerate Hopf bifurcation. Then generically there exist three open regions in the parameter space giving rise to two, one or no periodic orbit. A general classification of degenerate Hopf bifurcations is given by Takens [25] in terms of Birkboff normal forms. Another classification, in the context of imperfect bifurcation theory which emphasises a distinguished bifurcation parameter [10], has been presented by Golubitsky and Langford [9]. Whereas the dynamics of the Hopf bifurcation lies in a two-dimensional center mani- fold, a simple steady state bifurcation can be reduced to a one-dimensional system. Here the coexistence of two stable equilibria, often called bistability, plays a particular impor- tant role in a variety of real systems, for example in chemical reactors and combustion problems [6], in nonlinear electric circuits [24] and in passive optical systems [19]. Gener- ically it occurs in two parameter families of differential equations, and is closely related to the cusp of elementary catastrophe theory [22]. The dynamics associated with a ttopf or a steady state bifurcation is easily understood in principle, but interesting phenomena can occur if these two types of bifurcations coalesce. The generic situation for such a coalescence is of codimension two and has been discussed by Langford [15], Guckenheimer

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