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

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

H sNp 2 114 1 2 Figure

H sNp 2 114 1 2 Figure 4: Stability diagram and phase portraits for the degenerate Hopf bifurcation dH. 1 SLcl H Of' Figure 5: Stability diagram and phase portraits for the two cases of the Takens-Bogdanov bifurcation TB. They are distinguished by, respectively, a stable and an unstable limit cycle in region 3. replaced by H and SNp, and the role of S-points and A-points is taken over by A-points and limit cycles, respectively. A local unfolding is shown in Figure 4. TB: Takens-Bogdanov bifurcation. This occurs if an A-point has a nilpotent lineariza- tion. At the codimensiou 2 point a saddle node and a Hopf bifurcation with its period going to infinity coalesce. As a consequence the local unfolding contains 5'Na, H and SLa as subsidiary codimenslon 1 bifurcations. The stability diagram and the phase portraits for the two cases of TB are shown in Figure 5. P -t- SNs: coincident pitchfork and saddle node bifurcation. For the 3-d system this situation describes generic interactions between a Hopf and a saddle node bifurcation. The underlying normal form for the reduced (r, x)-system has been set up and classified by Guckenheimer [11,12]. At the bifurcation a pair of A-points and two S-points coalesce. There are essentially three different cases, all of which appear in the unfolding (9). For one of these the bifurcating A-point is a saddle while for the other two it is a focus that undergoes a ttopf bifurcation. The unfolding geometries corresponding to the latter two cases are shown i n Figure 6(a), (b). Observe the annihilation of the limit cycle in Figure 6(b) when one passes from region 4 to 5 or from 4' to 3. The loca~ normaJ form of [11,12] predicts here a limit cycle that grows towards infinity and then disappears. This artefact of the local normal form is resolved if the codimension 2 bifurcation is embedded in the unfolding of a higher singularity. In the case of our unfolding the limit cycle is destroyed

SLs(H} H.(SL s) SNs I SN~ (~6) 115 l[ I_"=-'I = _ > lff 2It 3k_~ , 5 ",,x_J {b) ! . I L,I, t 6 Figure 6: Two cases (a), (b) for generic interactions of a saddle node and a pitchfork. The limit cycle may be stable (phase portrait 4) or unstable (phase portrait 4'), depending on the relative position of the Hopf line H and the llne of global bifurcations (SLs in (a), G in (b)). The line G in (b) corresponds to the creation or annihilation of a limit cycle by means of some global mechanism which is not governed by the local normal form. SNp 2~~ 3 SLa Figure 7: Stability diagram and phase portraits for a degenerate saddle loop (dSLa).

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