Evolution and stability of the theoretically predicted families of periodic orbits in the N-body ring problem

We theoretically investigate the existence of families of periodic orbits in the planar N-body ring problem and we give a qualitative picture of the motion of a small particle. This study yields four families of periodic orbits which we also found numerically: two families of periodic orbits around the central body and two families around all the peripherals. These results are valid for different values of N and β. Also we investigate the evolution of simple periodic motions as well as their stability. We found stable and unstable orbits around the central body, around all the peripherals and around one or more peripherals which form rings of stability. Some families present other types of bifurcations, such as bifurcations of families of non-symmetric periodic orbits of the same period and period-doubling bifurcations. © ESO 2005

Infinite Feigenbaum sequences and spirals in the vicinity of the Lagrangian periodic solutions

We studied systematically cases of the families of non-symmetric periodic orbits in the planar restricted three-body problem. We took interesting information about the evolution, stability and termination of bifurcating families of various multiplicities. We found that the main families of simple non-symmetric periodic orbits present a similar dynamical structure and bifurcation pattern. As the Jacobi constant changes each branch of the characteristic of a main family spirals around a focal point-terminating point in x- at which the Jacobi constant is C∞ = 3 and their periodic orbits terminate at the corotation (at the Lagrangian point L4 or L5). As the family approaches asymptotically its termination point infinite changes of stability to instability and vice versa occur along its characteristic. Thus, infinite bifurcation points appear and each one of them produces infinite inverse Feigenbaum sequences. That is, every bifurcating family of a Feigenbaum sequence produces the same phenomenon and so on. Therefore, infinite spiral characteristics appear and each one of them generates infinite new inner spirals and so on. Each member of these infinite sets of the spirals reproduces a basic bifurcation pattern. Therefore, we have in general large unstable regions that generate large chaotic regions near the corotation points L4, L5, which are unstable. As C varies along the spiral characteristic of every bifurcating family, which approaches its focal point, infinite loops, one inside the other, surrounding the unstable triangular points L4 or L5 are formed on their orbits. So, each terminating point corresponds to an asymptotic non-symmetric periodic orbit that spirals into the corotation points L4, L5 with infinite period. This is a new mechanism that produces very large degree of stochasticity. These conclusions help us to comprehend better the motions around the points L4 and L5 of Lagrange. © 2010 Springer Science+Business Media B.V

Periodic Motions and their Stability in a N = ν + 1-body Regular Polygonal Configuration

We present a systematic investigation of the parametric evolution of both retrograde and direct families of periodic motions as well as their stability in the inner region of the peripheral primaries of the planar N-body regular polygonal configuration (ring model). In particular, we study the change of the bifurcation points as well as the change of the size and dynamical structure of the rings of stability for different values of the parameters ν = N-1 (number of peripheral primaries) and β (mass ratio). We find some types of bifurcations of families of periodic motions, namely period doubling pitchfork bifurcations, as well as bifurcations of symmetric and non-symmetric periodic orbits of the same period. For a given value of N - 1, the intervals Δx and ΔC of the rings of stability (where the periodic orbits are stable) of both retrograde and direct families increase with β increasing, while for a given value of β, the interval ΔC decreases with increasing N - 1. In general, it seems that the dynamical properties of the system depend on the ratio (N - 1)/β. The size of each ring of stability tends to zero as the ratio (N - 1)/β → ∞, that is, if N - 1→∞ or β → 0, the size of each ring of stability tends to zero (Δx → 0 and ΔC → 0) and, in general, the retrograde and direct families tend to disappear. This study gives us interesting information about the evolution of these two families and the changes of the bifurcation patterns since, for example, in some cases the stability index A oscillates between -1 ≤ A ≤ + 1. Each time the family becomes critically stable a new dynamical structure appears. The ratios of the Jacobian constant C between the successive critical points, Ci/Ci+1, tend to 1. All the above depend on the parameters N - 1, β and show changes in the topology of the phase space and in the dynamical properties of the system. © 2010 Springer Science+Business Media B.V

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 & 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. © 2011 World Scientific Publishing Company