Chains of rotational tori and filamentary structures close to high multiplicity periodic orbits in a 3D galactic potential

This paper discusses phase space structures encountered in the neighborhood of periodic orbits with high order multiplicity in a 3D autonomous Hamiltonian system with a potential of galactic type. We consider 4D spaces of section and we use the method of color and rotation [Patsis and Zachilas 1994] in order to visualize them. As examples we use the case of two orbits, one 2-periodic and one 7-periodic. We investigate the structure of multiple tori around them in the 4D surface of section and in addition we study the orbital behavior in the neighborhood of the corresponding simple unstable periodic orbits. By considering initially a few consequents in the neighborhood of the orbits in both cases we find a structure in the space of section, which is in direct correspondence with what is observed in a resonance zone of a 2D autonomous Hamiltonian system. However, in our 3D case we have instead of stability islands rotational tori, while the chaotic zone connecting the points of the unstable periodic orbit is replaced by filaments extending in 4D following a smooth color variation. For more intersections, the consequents of the orbit which started in the neighborhood of the unstable periodic orbit, diffuse in phase space and form a cloud that occupies a large volume surrounding the region containing the rotational tori. In this cloud the colors of the points are mixed. The same structures have been observed in the neighborhood of all m-periodic orbits we have examined in the system. This indicates a generic behavior.Comment: 12 pages,22 figures, Accepted for publication in the International Journal of Bifurcation and Chao

Instabilities and stickiness in a 3D rotating galactic potential

We study the dynamics in the neighborhood of simple and double unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. In order to visualize the four dimensional spaces of section we use the method of color and rotation. We investigate the structure of the invariant manifolds that we found in the neighborhood of simple and double unstable periodic orbits in the 4D spaces of section. We consider orbits in the neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis (the rotational axis of our system). Close to the transition points from stability to simple instability, in the neighborhood of the bifurcated simple unstable x1v2 periodic orbits we encounter the phenomenon of stickiness as the asymptotic curves of the unstable manifold surround regions of the phase space occupied by rotational tori existing in the region. For larger energies, away from the bifurcating point, the consequents of the chaotic orbits form clouds of points with mixing of color in their 4D representations. In the case of double instability, close to x1v2 orbits, we find clouds of points in the four dimensional spaces of section. However, in some cases of double unstable periodic orbits belonging to the z-axis family we can visualize the associated unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky to this surface for long times (of the order of a Hubble time or more). Among the orbits we studied we found those close to the double unstable orbits of the x1v2 family having the largest diffusion speed.Comment: 29pages, 25 figures, accepted for publication in the International Journal of Bifurcation and Chao

The structure and evolution of confined tori near a Hamiltonian Hopf Bifurcation

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis and Zachilas 1994]. We find that the consequents are contained in 2D "confined tori". Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.Comment: 10 pages, 14 figures, accepted for publication in the International Journal of Bifurcation and Chao