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

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

108 [11,12] and Keener

108 [11,12] and Keener [14]. It corresponds to a simultaneous Hopf and saddle node bifur- cation. The interaction of a cusp and a non-degenerate Hopf bifurcation is discussed by Langford [16,17] and applied to a chemical reaction in [23]. The subject of this paper is the local unfolding of a singularity in which a cusp (or hysteresis) and a degenerate ttopf bifurcation coalesce. This singularity has codimension four and contains Langford's case [16,17] as a subsidiary bifurcation, i. e., we recover all phenomena which he has observed in certain regions of the parameter space. In other regions the behaviour is substantially more complex, because there is now the possibility of two periodic orbits. This leads, for example, to the appearance of a torus with one period going to infinity as a subsidiary codimension one bifurcation. Also two nested tori, one stable and the other unstable may occur. Our analysis was motivated by a paper of Armbruster [1] who argued, guided by nu- merical results of Lugiato et al. [18], that the singularity described before might occur in the Maxwell-Bloch equations underlying optically bistable systems. By using the classi- fication of [2,3] he performed a singularity analysis of the generic bifurcation diagrams, that is, he investigated the stationary and periodic solution branches in the context of imperfect bifurcation theory [10]. In contrast to his work we were able to analyze the dynamical behavlour of the unfolded Poincare-Birkhoff normal form to some extent com- pletely, including local and global bifurcations to tori. Also we have explicitely calculated the physical parameter values for which the codimension-four bifurcation occurs in the Maxwell-Bloch equations. The results of this cMculation are summarized in the Appendix. In Section 2 the normal form corresponding to the singularity under consideration is established and simplified. Any truncated normal form corresponding to the coalescence of Hopf and steady state bifurcations, no matter how degenerate these are, possesses a St-symmetry that follows from the temporai translation invariance of the periodic orbits. As a consequence the dynamics can be described by two-dimensional phase portraits. In Section 3 we discuss the subsidiary singularities that occur in the unfolded normal form. A two-dimensional section through the stability diagram, together with the structurally stable phase portraits, is presented in Section 4. We also comment on possible phenomena which may occur if the normal form symmetry $1 is broken and report briefly about some numerical observations. 2 The normal form Consider a system of differential equations, /~ = f(u), (I) where ueR 3, f : 1~ 3 --~ R, 3 iS sufficiently smooth and f(0) = 0 so that u = 0 is an equilibrium. We assume that the linearizatlon L = duf(O) has a simple pair of purely imaginary eigenvalues =t:iw and a simple eigenvalue 0. With a linear change of coordinates and a rescaiing of time to make w = 1, the matrix L can be brought into the form (0 -1 0) L= 1 0 0 . (2) 0 0 0

109 Setting x = u3 and introducing polar coordinates, ul + iu2 = re i~, the'Poincare-Birkhoff normal form corresponding to L takes the form ÷ = rgl(r2,=) (3a) = g2(r2,x) (35) = l+g~(r~,=), (3c) where the g~ are smooth functions which vanish at the origin, and g2,=(0, 0) = 0. The significance of (3) is that there exists a sequence of near identity transformations which transforms (1) into (3) to arbitrarily high order, however, convergence is not assured. The S'l-symmetry mentioned in the Introduction corresponds here to the phase shift invariance ¢ --~ ¢ + ¢ of (3). As a consequence ¢ decouples from (r, x) so that the essential "normal form dynamics" is described by the two-dimensional (r, x)-system (3a,b) which possesses the reflection symmetry r --* -r. Our purpose is to describe a situation where both the steady state and the Hopf bifurcation become degenerate. This means that we have to impose the conditions in order that the saddle node degenerates to a cusp, and g2,==(0,0) = 0, (4a) g,.=(0,0) = 0, (4b) which induces a degeneracy in the Hopf bifurcation. In applications the system (1) or (3), together with the conditions (4), typically describes the flow in the center manifold of a degenerate equilibrium that occurs in a four parameter family of vector fields at an isolated point in the parameter space. The Taylor expansion of the (r, z) system reads ÷ = r [~=~ +,o,. ~ + ~=~ + q3x,~ + q.,x', + q,=~,.~ + qor" + 0(5)] (Sa) = ax 3 + br 2 + dxr 2 +plx 4 + qlx2r 2 + q2r 4 + 0(5), (5b) with certain coefficients a, b, c, d, P0 etc. We assume that a, b, c ~ 0. In order to simplify the system (5) we apply three successive near-identity transformations. The first step is to obtain pj = 0 (j = 0, 1,2). This is achieved by the following change of variables, where which transforms (5) into r --~ r(1 -pox/b),x ~ x + Az2,t ~ t(1 - Bz), A = p2[c - pl/a - poa/bc, B = m - pl/a, ÷ = r [cz 2 + q~zr ~ + q~z 4 + q~z~r ~ + q~6r 4 + 0(5)] (6a) = ax 3 + br 2 + exr 2 -]- q~x2r 2 + q~r 4 + 0(5). (6b) The q~ are new coefficients depending on those occuring in (5), and e is given by e = d+ (2- 3alc)po+ (31c-41a)bp,. (7)

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