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

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

110 To remove the terms

110 To remove the terms associated with the coefficients q~, q~, q~ in (6) we try the ansatz x --~ x + Ar 2 + BzZ, t --* t(1 - Cz2), with A, B, C yet undetermined. This produces again a system of the form of (6) with new coefficients q~'. For j = 1,3,4 they are given by q~t = q~ 4- (2c - 3a)A + b(3B 4- C) = 2cA c(C - 2B). The conditions q~t = q~ = q~ = 0 lead to an invertible linear system of equations. Thus, by making the proper choice of A, B, C, we may assume that these coefficients vanish. To remove the remaining quartic terms we apply to (6), with the q~ replaced by q~, the change of variables r --* r(1 4- Axr2),x ~ z + Bxr2,t ~ t(1 - Cr2), which yields once more a system of the form of (6) with new coefficients q~" given by q~!l = q~! = 0 for j = 1, 3, 4 and q;'+ b(B + C) = qi' + c(C - 2B) = q '+bA. As before we may choose A, B, C such that all coefficients q~!~ vanish. Thus the original system (5) has been transformed to the simpler system = r [cx + 0(5)] (sa) = ax 3 4- br 2 4- exr 2 4- O(5), (8b) with e given by (7). Assuming that a,b,c,e ~ 0, a simple rescaling of x, r, t changes these coefficients such that lal = Ibl = lel = 1 and c is transformed to c/a. A preliminary analysis indicates that the O(5)-terms in (8) can be removed to any desired order. We will return to this point elsewhere. If only the equilibria of (8) (and of its unfolding) are to be classified in the context of singularity theory, then all O(5)-terms can be made to vanish and it is also possible to achieve e = 0. This simplification was used by Armbruster [1]. His bifurcation diagrams have been derived from an unfolding of (8) with e = 0. In contrast, for the dynamics it is crucial that e is nonzero because otherwise the Hopf bifurcations in the unfolding of (8) (torus bifurcations in the original three-dimensional system) become highly degenerate. We are particularly interested in the case where a,e < 0, e > 0 because this occurs in the Maxwell-Bloch equations underlying passive optical systems (see Appendix). Since under the reflection x --+ -x (a, b, c, e) is transformed into (a, -b, c, e) the sign of b does

111 not matter. We choose b > 0. Then, after rescaling the variables and replacing c by -c with c > 0, a four parameter unfolding of (8) is given by ÷ = -cr(x 2 + 7x "F ,5) (9a) = -(x 3+r 2-xr 2+~x+a), (9b) where a, 8, 7, ~ are the unfolding parameters. In (9) we have neglected the O(5)-terms. In the next section we describe the subsidiary bifurcations organized by the unfolding (9) and in Section 4 we present some of the structurally stable phase portraits. The complete three-dimensional "normal form dynamics" is obtained simply by considering also variations of the phase according to (3c). When the normal form symmetry is broken, the dynamics becomes more intricate (See [16,23] and Section 4). 3 The subsidiary bifurcations In order to understand the stability diagram for the unfolding (9) we need to know the bifurcations of codimension smaller than four which occur on certain sets in the (c~, fl, 7, g)- space as subsidiary bifurcations of the normal form. Here and in the next section we regard (9) as a truly two-dimensional system with a reflection symmetry. We therefore use a terminology that is adapted to this kind of systems, but explain the meaning of the various bifurcations also for the full three-dimensional dynamics. Planar vector fields with a reflection symmetry r ~ -r possess two types of equilibria, which we will refer to as S-points: (r(t), a~(t)) ---- (0, xo) A-points: ~r(t), z(t)) = (r0, z0), r0 ~ 0. The S-points are true stationary solutions of the underlying three-dimensional system, however, the A-points are rotated in virtue of the phase variation (3c) and so correspond to periodic orbits. Typically they are produced at a pitchfork bifurcation from an S-point in the two-dimensional vector field. This pitchfork corresponds to a Hopf bifurcation in the three-dimensional system. An overview of the subsidiary local bifurcations of (9) is given in the subordination diagram of Figure 1. Here each of the rows i through 4 contains the bifurcations of codimension I through 4 as indicated in the column "cod". The arrows indicate the subordination structure: a singularity organizes anothe~ singularity of lower codimension if the latter can be reached from the former by a sequence of arrows. In Figure 1 we have confined ourselves to local bifurcations which are condensed in single equilibria. In addition to these also a number of global bifurcations of the saddle loop type occurs. In what follows we describe in some detail the bifurcations up to codimension two and comment briefly on those of codimension three. The presentation here is qualitative; a more comprehensive discussion including local normal forms and equations for the varieties in (a, ~,3,,6)-space where these singularities occur is in preparation.

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