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The Newton-Raphson basins of convergence, related to the equilibrium points, in the collinear restricted four-body problem with repulsive Manev potential are numerically investigated. We monitor the parametric evolution of the position as well as of the stability of the equilibrium points, as a function of the parameter e. The multivariate Newton-Raphson optimal method is used for revealing the basins of convergence, by classifying dense grids of initial conditions in several types of two-dimensional planes. We perform a systematic and thorough analysis in an attempt to understand how the parameter e affects the geometry as well as the basin entropy of the convergence regions. The convergence areas are related with the required number of iterations and also with the corresponding probability distributions. [ABSTRACT FROM AUTHOR]

In the present problem, we have studied the effect of small perturbations in the Coriolis and centrifugal forces on the locations and stability of libration points by taking the smaller primary as radiating oblate spheroid in the restricted four‐body problem. We have supposed that all the primaries are set in an equilateral triangle configuration, moving in circular orbit around their common center of mass. We have observed that the oblateness and radiation of the third primary, and the effect of the small perturbation in centrifugal force have substantial influences on the locations of libration points but the effect of the small perturbation in the Coriolis force has no impact on their locations. However, the stability of the libration points is highly influenced by the effect of the small perturbation in the Coriolis force. Furthermore, it is obtained that the effect of the small perturbation in the centrifugal force has a substantial influence on the regions of possible motion. In addition, we have also discussed how the oblateness and radiation pressure of the third primary, and the small perturbations in the Coriolis and centrifugal forces influence geometry of the basins of convergence by using multivariate version of the Newton–Raphson iterative scheme. [ABSTRACT FROM AUTHOR]

Highlights • The restricted four-body problem with pseudo-Newtonian potential is investigated. • The position and stability of the libration points are discussed. • The number of libration points depend on transition parameter. • The evolution of the region of possible motion are discussed. • The Newton-Raphson iterative scheme is used to unveil the basins of convergence. Abstract The dynamics of the pseudo-Newtonian restricted four-body problem has been studied in the present paper, where the primaries have equal masses. The parametric variation of the existence as well as the position of the libration points are determined, when the value of the transition parameter ϵ ∈(0, 1]. The stability of these libration points has also been discussed. Our study reveals that the Jacobi constant as well as transition parameter ϵ have substantial effect on the regions of possible motion, where the fourth body is free to move. The multivariate version of Newton-Raphson iterative scheme is introduced for determining the basins of attraction in the configuration (x, y) plane. A systematic numerical investigation is executed to reveal the influence of the transition parameter on the topology of the basins of convergence. In parallel, the required number of iterations is also noted to show its correlations to the corresponding basins of convergence. It is unveiled that the evolution of the attracting regions in the pseudo-Newtonian restricted four-body problem is a highly complicated yet worth studying problem. [ABSTRACT FROM AUTHOR]

In this paper, we present the analysis of the Newton–Raphson basins of convergence (NR-BoCs) linked with the relative equilibria (LPs) in the Eulerian configuration of a pseudo-Newtonian planar circular restricted problem of four bodies (SCCR4BP) with spinning primaries. In this configuration of the SCCR4BP, the primaries are laid out in a symmetrical collinear shape in which the two peripheral primaries are of equal mass. We derive the equations of motion by assuming that the system consists of three Kerr-like primaries to examine how the geometry of the basins of convergence (BoCs) associated with the LPs of SCCR4BP are affected by the first-order non-Newtonian contribution to the gravitational field. Additionally, we analyze the correlation among the regions of convergence, the required number of iterations, and the associated probability distributions. Finally, for the quantitative study of the basins of convergence, we measure the uncertainty of the respective basins via the basin entropy and the basin boundary entropy. • On the beyond-Newtonian collinear circular restricted (3 + 1)-body problem with spinning primaries is discussed. • The Newton–Raphson basins of convergence are discussed. • The basin entropy and basin boundary entropy are calculated to measure the uncertainty of the basins. [ABSTRACT FROM AUTHOR]

The present manuscript deals with the restricted four-body problem when all the primaries are oblate spheroids. This paper unveils the effect of oblateness on the existence, locations and stability of the libration points. It is assumed that three oblate primaries having equal masses are set in Lagrangian equilateral triangle configuration. It is observed that there exist ten libration points in configuration ( x , y ) -plane out of which four are collinear and six are non-collinear for oblateness parameter A ∈ [ 0 , 0 . 681949 ) . Further, for the oblateness parameter A ∈ ( 0 . 681949 , 1 ) , there exist only four libration points out of which two are collinear and two are non-collinear. Out-of-plane libration points are also investigated and it is found that in-plane and out-of-plane libration points are unstable. It is further observed that oblateness parameter has substantial effect on the regions of possible motion. Further, we have numerically investigated the Newton–Raphson basins of convergence associated with the libration points to unveil that how the oblateness parameter influences the domain of convergence. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the existence, locations and stability of the equilibrium points as well as zero velocity curves (ZVCs) under the effect of dissipative force (Stokes drag) in the restricted four-body problem (R4BP) with a variable mass. We have studied the dynamics of the infinitesimal body whose mass is variable and changing according to Jeans' law. We have determined the equations of motion of the infinitesimal body by using the space time transformation of Meshcherskii when the change in mass is non-isotropic. We have determined the locations of equilibrium points in the special case of non-isotropic variation of mass under the effect of the dissipative force and found that all the equilibrium points are non-collinear. The effect of parameter k (0 < k < 1) , which controls the dissipative force, on the locations of the libration points and their stability are investigated. It has been observed that the energy integral is time dependent due to the presence of the Stokes drag force. Further, it has been observed that all the equilibrium points are unstable for all values of the parameters used. • The effect of Stoke's drag in the circular restricted four-body problem with variable mass has been studied. • The locations of equilibrium points and their stability have been discussed. • Dissipative constant k has more impact on the locations of the equilibrium points. • The effect of Stokes drag also has been studied by the zero velocity curves on the regions of possible motion. • The effects of variable mass and Stokes drag have significant impacts on the regions of possible motion. [ABSTRACT FROM AUTHOR]