Families without perturbation

In order to understand the influence of the warp, we first look at the model without deformation ($w=0$) which is completely axisymmetric. The rotation of the potential is set to zero ($\Omega_p =0$).

Below the orbit, families are shown with their initial starting point in the $H-p_z(0)$ diagram. In Fig. 2, bottom, we recognize the circular orbit family (horizontal line at $p_z(0)=0$). The stability indices for this family are traced at the top of Fig 2. In this particular case, $b_1$ indicates the stability in the galaxy plane while $b_2$ corresponds to the stability transverse to it. Both indices remain in the interval $[-2,+2]$, insuring the stability for this family.

At $H=-0.0701$, $b_1=+2$, $\nabla T$ is also degenerate with two eigenvalues equal to $-1$. A bifurcation occurs through period doubling. This bifurcation coincides with the resonance between the radial frequency $\kappa$ and the circular frequency $\Omega$ ( $2\kappa=3\,\Omega$).

At $H=-0.0748$, $-0.0566$, $-0.0451$, $-0.0345$, and $-0.0236$, $b_2=+2$ generating transverse bifurcations through period doubling. These bifurcations coincide with the resonances between the transverse frequency $\nu$ and the circular frequency $\Omega$ ( $2\nu=(2k+1)\,\Omega$, $k=1,\ldots 5$). Orbit families from period doubling bifurcations will not be discussed further in this paper.

Transverse bifurcations keeping the same period occur at $H=-0.0632$, $-0.0508$, $-0.0396$, $-0.0294$, and $-0.0158$. In this case $b_2=-2$ and $\nu = k\, \Omega$, $k=2,\ldots 6$. The two latter families have a non zero $p_z(0)$ initial velocity and also oscillate round the $z=0$ plane crossing it respectively $2k$ and $2(k+1)$ times per period. All these sub-families are marginally stable with $b_2=-2$ due to the axisymmetry of the potential.

Figure 2: $H$-$p_z(0)$ phase space for $w=0$.