Sitnikov problem

Abstract: We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” z0 varies within a finite interval (while z 0 tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincare maps, we chart the motion near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity.

Abstract: We present results about the stability of vertical motion and its bifurcations into families of 3-dimensional (3D) periodic orbits in the Sitnikov restricted N-body problem. In particular, we consider ν = N − 1 equal mass primary bodies which rotate on a circle, while the Nth body (of negligible mass) moves perpendicularly to the plane of the primaries. Thus, we extend previous work on the 4-body Sitnikov problem to the N-body case, with N = 5, 9, 15, 25 and beyond. We find, for all cases we have considered with N ≥ 4, that the Sitnikov family has only one stability interval (on the z-axis), unlike the N = 3 case where there is an infinity of such intervals. We also show that for N = 5, 9, 15, 25 there are, respectively, 14, 16, 18, 20 critical Sitnikov periodic orbits from which 3D families (no longer rectilinear) bifurcate. We have also studied the physically interesting question of the extent of bounded dynamics away from the z-axis, taking initial conditions on x, y planes, at constant z(0) = z 0 values, where z 0 lies within the interval of stable rectilinear motions. We performed a similar study of the dynamics near some members of 3D families of periodic solutions and found, on suitably chosen Poincare surfaces of section, “islands” of ordered motion, while away from them most orbits become chaotic and eventually escape to infinity. Finally, we solve the equations of motion of a small mass in the presence of a uniform rotating ring. Studying the stability of the vertical orbits in that case, we again discover a single stability interval, which, as N grows, tends to coincide with the stability interval of the N-body problem, when the values of the density and radius of the ring equal those of the corresponding system of N − 1 primary masses.

Abstract: The main goal of this paper is to study analytically the families of symmetric periodic orbits of the elliptic Sitnikov problem for all values of the eccentricity in the interval (0, 1), providing qualitative and quantitative information on the bifurcation diagram of such families of periodic orbits. The basic tool for proving our results is the global continuation method of the zeros of a function depending on one parameter provided by Leray and Schauder and based in the Brouwer degree.

Abstract: In this paper by means of a Poincare map, we prove the existence of symmetric periodic orbits of the elliptic Sitnikov problem. Furthermore, using the presence of the Bernoulli shift as a subsystem of that Poincare map, we prove that not all the periodic orbits of the Sitnikov problem are symmetric periodic orbits.