Numerical method

The periodic orbits are found by numerically determining the fixed points of the 4D Poincaré map ($T$) at $y=0$, $\dot x <0$ generated by the equations of motion (see, e.g., Pfenniger & Friedli [1993]). The algorithm uses in particular the method proposed by Hénon ([1982]) giving the intersection between an orbit and a surface, and a least squares stabilized Newton-Raphson root-finding procedure.

The stability of the orbits are determined by the eigenvalues of the Jacobian of the Poincaré map ($\nabla T$). Since the system is Hamiltonian and the motion is described by real numbers, the four eigenvalues of $\nabla T$ occur by conjugate and inverse pairs and it is possible to condense the information with two stability indexes $b_1$, $b_2$:

$$b_i = -(\lambda_i + \lambda_i^{-1}), \quad i=1,2, \\$$ (6)

where $\lambda_1$ and $\lambda_2$ represent a pair of reciprocal eigenvalues. A periodic orbit is stable only when $b_1$ and $b_2$ are real and $\vert b_1\vert$,$\vert b_2\vert<2$. It is unstable in all other cases. If $\vert b_1\vert=2$ or $\vert b_2\vert=2$, or if $\vert b_1\vert=\vert b_2\vert$, at least two eigenvalues are equal, $\nabla T$ is degenerate and eventually allows a bifurcation. For a more complete description of the instability cases, see Pfenniger & Friedli ([1993]).

In this work, we will focus our investigations on orbits following the potential disk. Hence, we choose initial conditions with a starting position on the line of nodes, here equivalent to the $y$-axis. We concentrate on the main orbit families keeping the symmetry of the potential, thus the initial velocity is perpendicular to it ($p_y(0)=0$) and points to arbitrary negative values of $x$ ($p_x(0)<0$). In this way, the free parameters in the initial conditions are the Jacobi constant ($\approx$ energy, $H$), the position $y(0)$ along the $y$ axis, and the velocity component $p_z(0)$ along the $z$ direction. This latter parameter will be small compared to $p_x(0)$, which avoids a situation where orbits stray too far from the disk and penetrate negative density regions. This point has been systematically checked because, a priori orbits may explore regions far from the expected ones.