On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries

Abstract

The existence of new equilibrium points is established in the restricted three-body problem with equal prolate primaries. These are located on the Z-axis above and below the inner Eulerian equilibrium point L 1 and give rise to a new type of straight-line periodic oscillations, different from the well known Sitnikov motions. Using the stability properties of these oscillations, bifurcation points are found at which new types of families of 3D periodic orbits branch out of the Z-axis consisting of orbits located entirely above or below the orbital plane of the primaries. Several of the bifurcating families are continued numerically and typical member orbits are illustrated.

The effect of zonal harmonic coefficients in the framework of the restricted three-body problem

“…Finally, some examples showing that there is no influence for J 4 on the range of stability for some planets systems as Earth-Moon, Saturn-Phoebe and Uranus-Caliban systems. Furthermore, the existence of new equilibrium points for the restricted three-body problem with equal prolate primaries is stated in Douskos et al (2012). It was found that these points are located on the Z-axis above and below the inner Eulerian equilibrium point L 1 as well as a new type of straight-line periodic oscillations, different from the well-known Sitnikov motions.…”

Divulging the effect of small perturbations in the Coriolis and centrifugal forces in the photogravitational version of autonomous restricted four‐body problem with oblate primary

“…In the present section, we use the well‐known multivariate version of the Newton–Raphson method to discuss the convergency of basins of attraction, which is a fast (converges quadratically), simple, and accurate computational tool. In this section, we shall use the computational method applied by Croustalloudi & Kalvouridis (), Douskos et al (), and Zotos (). The Newton–Raphson process starts with the given initial approximation ( x 0 , y 0 ) on the plane and stops when the libration point is found with a predetermined accuracy.…”

On the Newton–Raphson basins of convergence associated with the libration points in the axisymmetric restricted five-body problem: The concave configuration

“…Thus, the Newton-Raphson iterative scheme to determine the basins of convergence associated with the libration points, provides intrinsic properties of the dynamical system. A series of literature is available which deals with the basins of convergence associated with the libration points in various type of dynamical system such as the restricted three-body problem (e.g., [29], [24]), the restricted four-body problem with and without various perturbations (e.g., [15], [8], [9], [20,21], [30,31], [22]), the restricted five-body problem (e.g., [34]), the Hill's problem (e.g., [10], [33]), pseudo-Newtonian three and four body problem (e.g., [32], [25]), the Copenhagen problem (e.g., [35], [23]), or even the Sitnikov Problem in three and four body problem (e.g., [11], [36,37]). The present paper is described with following structure: the basic properties of the model of axisymmetric five-body problem are presented in Sect.2.…”