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

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

112 \ 1 / C P + SNs dP

112 \ 1 / C P + SNs dP dH TB 2 Figure 1: Subordination diagram of the local singularities organized by the unfolding (9). Codimension 1 The unfolding (9) exhibits the following local singularities of codlmension 1 as subsidiary bifurcations: SNs: symmetric saddle node giving rise to the coalescence of two S-points. For the 3-d system this corresponds to a generic saddle node. SiVa: asymmetric saddle node involving A-points. Its meaning for the 3-d system is that of a saddle node for periodic orbits. P: pitchfork bifurcation, i. e. , a pair of A-points (with opposite r0-values) is created from a S-point. It corresponds to a Hopf bifurcation in the 3-d system. //: Hopf bifurcation at an A-point. For the 3-d system it is associated with the formation of an invariant torus from a periodic orbit. In addition to the tIopf bifurcation, limit cycles in the (r, z)-plane are also created at two types of global bifurcations. These are: SL~,: homoclinic orbit of an A-point of the saddle type [13]. SLs: heteroclinic orbit in {r # 0} connecting two S-points which are both saddles [11,12]. The sequence of phase portraits in Figure 2(a) and (b) appears when one varies a parameter in a vector field through a critical value at which, respectively, a SL= and a SLs bifurcation takes place. For the cases shown in the figure a stable limit cycle is created/annihilated at the bifurcation. Analogeous versions with unstable limit cycles also can appear in the unfolding (9). For the 3-d system both global bifurcations give rise to an invariant torus with one of the two periods diverging as one approaches the bifurcation point. For SL,z and SLs the torus evolves into the intersection of the stable and unstable manifold of a periodic orbit and of two different equilibria, respectively. In the former case the toroidal structure is still visible whereas in the latter case the torus degenerates into the union of a sphere-like surface and a single trajectory. Also connected with the birth of limit cycles is the following global bifurcation: cod 4

113 SLot x SLs Figure 2: Phase portraits in the (r > 0, x)-half plane near the global bifurcations (a): SL,, and (b): SL,,,. SNo I 3j 3 ] "' p 2 r r -- ~- ~X Figure 3: Stability diagram and phase portraits in the (r > 0, x)-half plane for the degen- erate pitchfork dP. SNp: saddle node for periodic orbits. A pair of limit cycles, one of them stable and the other unstable, coalesces and disappears, analogeously to saddle node phenomena involving equilibria. Codimension 2 The local singularities are: C: cusp bifurcation, i. e. three S-points coalesce. In a two dimensional unfolding plane one encounters two lines of saddle nodes SN~ which meet in a cusp point. This singularity organizes the transition between monotonic and bistable behaviour of steady state branches [22]. In the context of imperfect bifurcation theory it is also called a hysteresis point [10]. dP: degenerate pitchfork. Two pairs of A-points are created from a S-point, i. e., in the 3-d system there exist two periodic orbits near this codimension two bifurcation. In the plane of unfolding parameters a SNa-line terminates with a second order contact on a P-line as shown in Figure 3. Observe that for parameters in region 1 the S-point is here a (stable) node. There is also another version of dP where it is a saddle. Both versions occur in (9). dH: degenerate Hopf bifurcation. This is analogeous to dP if there P and SNa are (a)

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