If the warp has only a slight influence on the shape of the orbits, it is important to look carefully at the behavior of the bifurcations in presence of the perturbation. Fig. 5 and Fig. 8 present a magnification of the - phase space around the first () and second () bifurcations. Here the periodic orbits are computed with in order to increase the effect of the warp. The physical correspondent would be a galaxy with a warp twice as high as that observed in the Milky Way at a radius of .
Except for the weak decreasing of with respect to , the type of the first bifurcation (around ) remains similar to the one without perturbation (pitchfork). The two new generated families ( and ) are both stable. They appear at the same energy as two other unstable families, namely and . These are not seen in Fig. 5 because their projection merges with the family. They are distinguished from it by a non zero and . Projections of families and are plotted in Fig. 7.
In order to estimate the importance of orbits associated with a stable periodic orbit, we have computed the surface of the section near the bifurcation (). Moreover, it allows us to know the number of effective integrals of motion in this specific region of phase space. In Fig. 6 the stable family is represented by the cross near the center (, ) and is surrounded by quasi periodic families. The thinness of these curves indicates that the motion is well decoupled in and a third integral exists. At and we find respectively the and families. The upper and lower panels show details of the section around these two families. Very thin stability islands exist. For comparison, the width in of the separatrix (squares points) is about times smaller than the width between families and . This gives an indication of the phase space occupied by orbits associated with these families. The unstable and families appear on the separatrix, at and . Outside the separatrix, the invariant curves are regular until where the motion in begins to be strongly coupled to the motion in and .
The behavior of the second bifurcation around is quite different. Fig. 8 reveals that the family is in fact the extension of the main family . This arises because the perturbation favors a higher . The family is also stable. At the families and are created simultaneously. They are respectively stable and unstable. The whole bifurcation forms a pitchfork with a symmetry breaking. The shapes of three of the four families involved in this bifurcation are presented in Fig. 10. The section after the bifurcations is shown in Fig. 9 (). The stable family appears at and is surrounded by a set of quasi periodic orbits embedded in the separatrix. The unstable family belongs to the separatrix at . The family is located at and its associated orbits occupy the crescent-shaped region between the separatrix. Outside, the invariant curves are regular until where the motion in is not longer decoupled.
Our study of the bifurcation is limited to the two first bifurcations, and . The behavior of the following bifurcations are similar to the first two. The bifurcation with an odd (resp. even) is of the same type as (resp. ). This difference is related to the symmetry of the families which depends on the parity of .
In summary, the warp does not destroy the strength of the main family, which at any possesses a comfortable surrounding stable region. For an odd , the effect of the warp is : (i) to change the stability of subfamilies and (ii) to increase the phase space allowed to the associated families. For an even , the subfamilies are quasi unchanged, preserving a very thin phase space around the stable subfamilies .