next up previous
Next: Influence of rotation Up: Families of periodic orbits Previous: Influence of the warp

Surfaces of section around bifurcations

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 $H$-$p_z(0)$ phase space around the first ($r1$) and second ($r2$) bifurcations. Here the periodic orbits are computed with $w=0.01$ 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 $30\,\textrm{kpc}$.

Except for the weak decreasing of $p_z(0)$ with respect to $H$, the type of the first bifurcation (around $H=-0.0633$) remains similar to the one without perturbation (pitchfork). The two new generated families ($s2+$ and $s2-$) are both stable. They appear at the same energy as two other unstable families, namely $z2+$ and $z2-$. These are not seen in Fig. 5 because their projection merges with the $p2$ family. They are distinguished from it by a non zero $z(0)$ and $p_y(0)$. Projections of families $s2+$ and $s2-$ 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 $(z,p_z)$ near the bifurcation ($H=-0.06$). 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 $p2$ family is represented by the cross near the center ($z=0$, $p_z=-0.0058$) and is surrounded by quasi periodic families. The thinness of these curves indicates that the motion is well decoupled in $z$ and a third integral exists. At $p_z=0.031,z=0$ and $p_z=-0.043,z=0$ we find respectively the $s2+$ and $s2-$ 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 $p_z$ of the separatrix (squares points) is about $1/100$ times smaller than the width between families $s2+$ and $s2-$. This gives an indication of the phase space occupied by orbits associated with these families. The unstable $z2+$ and $z2-$ families appear on the separatrix, at $p_z=-0.0054,z=0.427$ and $p_z=-0.0054,z=-0.427$. Outside the separatrix, the invariant curves are regular until $p_z= 0.06$ where the motion in $z$ begins to be strongly coupled to the motion in $x$ and $y$.

Figure 5: $H$-$p_z(0)$ and $H$-$y(0)$ phase space around the first bifurcation ($w=0.01$).
\resizebox{\hsize}{!}{\includegraphics{r1.eps}}

Figure 6: Section $(z,p_z)$ at $H=-0.06$, near the bifurcation $r1$. The crosses represent the three stable periodic families $s2+$ (top), $p2$ (middle) and $s2-$ (bottom) and the two unstable $sz+$ (right) and $sz-$ (left). The upper and lower panels show a zoom of the section around $s2+$ and $s2-$. In this panels the separatrix is marked with squares while in the middle it is marked with a bold line. The crosses indicate the position of the periodic families.
\resizebox{\hsize}{!}{\includegraphics{r1susec.eps}}

Figure 7: Shape of three orbits linked to the $r1$ bifurcation ($H=-0.06$). The dot indicates the starting point of the computation and the arrow shows the direction of the corresponding initial velocity.
\resizebox{\hsize}{!}{\includegraphics{orbit.r1.eps}}

The behavior of the second bifurcation $r2$ around $H=-0.0509$ is quite different. Fig. 8 reveals that the family $s3+$ is in fact the extension of the main family $p2$. This arises because the perturbation favors a higher $p_z$. The $s3+$ family is also stable. At $H=-0.0508$ the families $p3$ and $s3-$ 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 ($H=-0.05$). The stable $p3$ family appears at $p_z=-0.009$ and is surrounded by a set of quasi periodic orbits embedded in the separatrix. The unstable $s3-$ family belongs to the separatrix at $p_z=-0.023$. The $s3+$ family is located at $p_z=0.006$ and its associated orbits occupy the crescent-shaped region between the separatrix. Outside, the invariant curves are regular until $p_z=-0.03$ where the motion in $z$ is not longer decoupled.

Figure 8: $H$-$p_z(0)$ and $H$-$y(0)$ phase space around the second bifurcation ($w=0.01$).
\resizebox{\hsize}{!}{\includegraphics{r2.eps}}

Figure 9: Section $(z,p_z)$ at $H=-0.05$, after the bifurcation $r2$. The separatrix is marked with a bold line. The crosses indicate the position of the three periodic families $s3+$ stable (top), $p3$ stable (middle) and $s3-$ unstable (bottom)
\resizebox{\hsize}{!}{\includegraphics{r2_2susec.eps}}

Figure 10: Shape of three among the four orbits linked to the $r2$ bifurcation ($H=-0.0509$). The dot indicates the starting point of the computation and the arrow shows the direction of the corresponding initial velocity.
\resizebox{\hsize}{!}{\includegraphics{orbit.r2.eps}}

Looking at Fig. 6 and 9, one can expect a chaotic region around the separatrix. Due to the thinness of the region around the unstable points, it is very difficult to find it numerically.

Our study of the bifurcation is limited to the two first bifurcations, $r1$ and $r2$. The behavior of the following bifurcations are similar to the first two. The $rk$ bifurcation with an odd (resp. even) $k$ is of the same type as $r1$ (resp. $r2$). This difference is related to the symmetry of the families which depends on the parity of $k$.

In summary, the warp does not destroy the strength of the main family, which at any $H$ possesses a comfortable surrounding stable region. For an odd $k$, the effect of the warp is : (i) to change the stability of $sk-$ subfamilies and (ii) to increase the phase space allowed to the associated $sk+$ families. For an even $k$, the subfamilies are quasi unchanged, preserving a very thin phase space around the stable subfamilies $sk\pm$.


next up previous
Next: Influence of rotation Up: Families of periodic orbits Previous: Influence of the warp
2002-03-16