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

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

116 by one of the two

116 by one of the two saddle loop bifurcations SLs or SL,~ (see Section 4). The following global codimension 2 bifurcation occurs in the unfolding (9): dSLa: degenerate version of SLa. When a homoclinic orbit occurs ~t a saddle whose linearization has a vanishing trace one can infer the existence of a curve of SNp-points terminating on a line of SL=-points in a generic two parameter family. The stability diagram near the point of coincidence of SLp and SLy, is shown in Figure 7. The situation is similar to dH (Figure 3), however, SNp teminates on SLa in a flat manner in contrast to the quadratic tangency of SNp and H in the unfolding of dH. A few further codimension 2 bifurcations may occur in the unfolding (9). These are essentially combinations of local and global phenomena, i. e., a closed loop occurs at a degenerate equilibrium. In the stability diagram such a situation manifests itself in a line of saddle loops that terminates on a line of local bifurcations (P or SN,,). Codimension 3 We describe briefly the local bifurcations of codimension 3 which are summarized in Fig- ure 1. Among these P + C and dTB are considered in [16,17,22] and [5], respectively. The remaining two have not been discussed so far. P + C: coincidence of a pitchfork and a cusp for S-points. For the 3-d system this corresponds to the interaction of a cusp or a hysteresis and a non-degenerate Hopf bifur- cation. dTB: degenerate Takens-Bogdanov bifurcation. This is a degenerate version of TB that produces two limit cycles. The local unfolding contains both dH and dSL,~ as sub- sidiary codimension 2 bifurcations. SNo + dP: coincident saddle node and degenerate pitchfork bifurcations. Besides the obvious local codimension 2 bifurcations P + SNs and dP this singularity also organizes TB. SN8 + P ÷ dH: coincidence of SNs + P and dH. An isolated point on the Hopf line in the stability diagram of SNs + P (see Figure 6) may correspond to a degenerate Hopf bifurcation. When a third parameter is varied this rill-point can move towards the SNs + P-point and then disappear. 4 A two-dimensional section through the stability diagram The most convenient way for presenting the stability diagram for the unfolding (9) is in terms of sections through the (a, 5)-plane. First the (/~,7)-plane is divided into regions that are bounded by curves along which codimension three bifurcations or transversal crossings of lower codimension bifurcations occur. Then, for generic (/~, 7) in each of these regions, the bifurcations of codimension one and two are displayed in a section through the (a, 5)-plane. One of these sections, which covers most of the phenomena that occur, is shown in Figure 8. The unfolding organizes 22 different structurally stable phase portraits. Among these 15 occur in the regions labelled in Figure 8. They are sketched in Figure 9. Some of them (phase portraits 1, 2, 5, 9, 10, 11, 13, 14) occur already in the lower codimension bifurcation P + C considered by Langford [16,17].

iii 1 SLo/" SNo ,y X 117 i 11 21 . H l:ivb 10 7//" 3 ~SN f'v / P SNs t_-o f 9 SNs Figure 8: A (a, 6)-section through the stability diagram of the unfolding (9). Local and global bifurcations are distinguished by solid and dashed lines. The dashed-dotted line SNp corresponds to a line of saddle nodes for periodic orbits. Points i, ii, etc. are codi- mension-two points. The phase portraits which occur in regions 1, 2, 3, ... are sketched in Figure 9. On the left and right SNs-lines in Figure 8 the two left and right S-points organized by (9) coalesce, respectively. The cusp point, where all three S-points coincide, is generically not visible in the (a, $)-sections, but it is easy to identify the other codimension 2 points. The points labelled i, ii, iii and v are points of the types dP, dH, TB and dSLa. The two points iva and ivb are SN, + P-points corresponding to the cases shown in Figure 6(a) and (b), respectively. Observe that the artefact of the unspecified llne G of global bifurcations in Figure 6(b) is now resolved. Its role is taken over by the line SLa along which the phase portrait exhibits a trajectory connecting the intermediate and the right S-points which are both saddles. Let us now return to the full 3-d system, that is, the unfolding (9) is supplemented by the phase evolution (3c). The augmented system possesses a Sl-symmetry (¢ --* ¢ + ¢) which is reflected by the fact that the phase decouples from (r,x). However, this S1- symmetry is a normal form symmetry only and convergence of the transformation that brings (1) to the form (3) is not guaranteed. As a consequence, the Sl-symmetry must be broken at some level of the perturbation expansion. The question arises, to which 1 iva

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