The Poincaré section method shows that there are 2-D invariant tori and invariant curves (islands) around stable periodic orbits that bound to the surface of 3-D tori. The tube orbits preserve the direction of their circulation around either the long or short axis as appeared in the triaxial potential, even when every initial condition leads to different orientations. Conversely, a few chaotic trajectories are found with a rotated triaxial halo when (ϕ, q z)= (90, 1.5). The inspection of the maximal asymptotic Lyapunov (λ) exponent shows that the dynamical behaviors of the selected orbits are nearly the same as the regular orbits with two degrees of freedom, both in axisymmetric and triaxial when (ϕ, q z)= (0,0). We analysed the geometry of the dynamics and applied the Poincaré section method for comparing the influence of different birth velocity distributions. These isolated NSs are evolved under smooth, time-independent, 3-D gravitational potentials, axisymmetric and with a triaxial dark halo. We describe the dynamical behavior of isolated old (≥ 1 Gyr) objects-like Neutron Stars (NSs).
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