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The research presented in this work focuses on investigating the characteristics of equilibrium points within the context of the photogravitational restricted three-body problem. Specifically, the study considers the influences of albedo effect, a straight segment, radiation pressure, as well as small perturbations in the Coriolis and centrifugal forces. The dynamical model proposed in this study reveals the presence of five equilibrium points. Among these, three are collinear with the centers of the massive bodies, while the remaining two are non-collinear. By examining various parameters involved in the system, the effect on the position of the equilibrium points is explored through numerical and graphical analysis. These investigations provide valuable insights into how the positions of these points are affected by changes in the parameters. Another significant aspect addressed in this research is the stability of the equilibrium points. The study finds that the collinear equilibrium points are unstable, irrespective of the parameters involved. On the other hand, the non-collinear points exhibit stability within a specific range of the mass parameter. The precise range of stability depends on the values of the other parameters present in the system. Additionally, the research delves into the study of the permissible and forbidden regions of motion for an infinitesimal body within the system. This analysis is conducted by considering different values of the Jacobian constant and it suggests that the Jacobian constant significantly influences the dynamics of the system and has a direct impact on the permissible regions of motion of the infinitesimal body. [ABSTRACT FROM AUTHOR]

In non-linear stability analysis, the effects of non-linear terms in the equations of motion are considered. The presence of these non-linear terms can cause the results of linear stability to change dramatically. Therefore, the Arnold–Moser theorem (Kolmogorov–Arnold–Moser theory) has been used to study the non-linear stability of the equilibrium point in the planar Robe's restricted three-body problem when the second primary is a finite straight segment of length . The density parameter is considered zero. The equilibrium point has been found to be stable in non-linear sense for all mass ratios in the range of linear stability except possibly for six critical values of the mass ratio . The critical values depend on the length parameter , which shows that the length parameter has significant effect on the non-linear stability of the equilibrium point. The obtained results are applied to predict the stability of equilibrium point for Jupiter–Amalthea system. It is observed that the equilibrium point is unstable for Jupiter‒Amalthea system. [ABSTRACT FROM AUTHOR]

In the framework of axisymmetric problem of restricted five bodies with variable mass, our goal is to study the existence and stability of libration points. In the axisymmetric restricted five-body problem, we have assumed that the mass of the fifth particle varies according to Jeans' law. We have further supposed that the four bodies having masses and (with ) form an axisymmetric concave configuration. The equations of motion of a test particle of infinitesimal mass have been illustrated, which is similar to the axisymmetric restricted five-body problem when the problem of variable mass evolves with the problem of constant mass. In this paper, we have determined in-plane as well as out-of-plane libration points along with their stability. Further, we have shown the regions of motion where the fifth body can move freely. We have also shown that the angle parameters and and the parameters, arising due to variation of mass and , affect the existence and number of these libration points. Moreover, the bivariate version of the Newton-Raphson iterative scheme is applied in an attempt to unveil the analysis of the basins of convergence linked with the libration points as a function of the mass parameter. [ABSTRACT FROM AUTHOR]

The belief that celestial bodies as perfect spheres results in certain idealistic conditions because most often they are in irregular shapes. With this in mind, we conduct an analysis to study the restricted three-body problem by taking the primaries as oblate spheroids. The modified mean motion expression [1] is used by incorporating the secular perturbation effect due to oblateness of the primaries on mean anomaly, argument of perigee and right ascension of ascending node. The model possesses five equilibrium points, which are subsequently affected by the oblateness parameters. Furthermore, we have studied the stability of the equilibrium points and it is observed that the collinear equilibrium points are always unstable. However, the non-collinear equilibrium points are stable for some combinations of the involved parameters. We have also plotted the zero velocity curves of the infinitesimal body for different values of the Jacobian constant and oblateness parameter. It has been observed that the value of the Jacobian constant has been playing a vital role in obtaining the permissible regions of motion of the infinitesimal body. Further, the results obtained of the study are applied to study the motion of a satellite in the Saturn–Titan system. [ABSTRACT FROM AUTHOR]

The present study investigates the combined effects of the oblateness and straight segment on the positions and linear stability of the equilibrium points in the restricted 2 + 2 body problem. The present model holds fourteen equilibrium points, out of which six are collinear with the centers of the primaries and the rest are non-collinear. It has been observed that the positions of all the equilibrium points are subsequently affected by the oblateness and length of the primary bodies. The linear stability of the equilibrium points is also presented by slightly perturbing the position of the equilibrium points. It is observed that for a considered set of parameters, all the fourteen equilibrium points are unstable. An application of the present model is also studied, for which the position and stability of the equilibrium points are investigated for the Earth-22 Kalliope-dual satellite system. It has been observed that for this system, all the equilibrium points are unstable except for two non-collinear equilibrium points that are found to be stable. [ABSTRACT FROM AUTHOR]

In the present paper, a study on the beyond-Newtonian collinear circular restricted (3 + 1) -body problem with spinning primaries is presented. In the setup of the collinear circular restricted four-body problem (CCR4BP), the three primaries are situated in the collinear central configuration where the masses of two peripheral primaries are equal. We have first derived the equations of motion for a system composed of three Kerr-like primaries. Further, we have unveiled how the existence, evolution, and stability of the libration points in the CCR4BP with beyond-Newtonian first-order correction terms and spin of the primaries depend on the mass parameter β and control parameter ε intending to manage the contribution of the beyond-Newtonian terms. It is observed that equilibria of the system are always even and varying from six to eighteen. [ABSTRACT FROM AUTHOR]

The fractal basins of convergence (BoC) linked with the equilibrium points are explored in the (N + 1)− body ring problem under the effect of small perturbations in the Coriolis and centrifugal forces when the bodies are sources of radiation. The evolution of the positions and the stability of the libration points and the possible regions of motion are determined as the function of mass parameter, the Coriolis, and centrifugal and radiation parameters. In addition, the parametric variation of the effect of the small perturbations in the Coriolis and centrifugal forces on the stability of the equilibrium points are also illustrated numerically. The multivariate version of the Newton–Raphson(NR) iterative method is used to analyze the effect of the radiation parameters and mass parameter on the topology of the BoC. [ABSTRACT FROM AUTHOR]

In the present paper, an effort to generalize the restricted problem of 2 + 2 bodies has been made by considering the primary bodies m1 and m2 to be a point mass and straight segment of length 2l, respectively. The motion of the two infinitesimal bodies mi, i = 3, 4 has been investigated, in which each mi, i = 3, 4 is moving in the gravitational field of mj, j = 1, 2, 3, 4 with i ≠ j. For the present problem, 14 equilibrium points are obtained, of which 6 are collinear and the rest are noncollinear. The length parameter l has a subsequent effect on the location of all the equilibrium points. Systematic stability analysis has been carried out by using the theory of perturbation. All the equilibrium points are found to be unstable. [ABSTRACT FROM AUTHOR]

This article aims to numerically investigate the orbital dynamics of the planar circular‐restricted three‐body problem, where the two primary bodies have equal masses and an oblate spheroid shape. By numerically integrating several large sets of initial conditions of both retrograde and prograde orbits, we classify them into three main categories: (a) bounded (regular or chaotic), (b) escaping, and (c) collision orbits. We reveal the motion of the test particle by presenting color‐coded diagrams, where the initial conditions are mapped to the orbit type and studied as a function of the total orbital energy (or equivalently of the Jacobi constant) as well as of the oblateness coefficient. Our numerical results are compared with related previous ones, corresponding to the classical version of the planar‐restricted three‐body problem. The presented outcomes have practical physical applications as they can be used for modeling the motion of a massless test particle in the gravitational field of a binary system of two oblate exoplanets. [ABSTRACT FROM AUTHOR]

In this paper we numerically explore the convergence properties of the circular restricted problem of three primary bodies, when an additional repulsive quasi-homogeneous Manev-type correction term is added to the effective potential. The Newton-Raphson iterative scheme is used for revealing the basins of convergence and their respective fractal basin boundaries on the complex plane. A thorough and systematic analysis is conducted in an attempt to determine the influence of the Manev parameter on the convergence properties of the system. Furthermore, the roots (numerical attractors) of the system and several other quantities (such as the required number of iterations, the percentages of the attracting regions and the basin entropy of the convergence diagrams) are monitored as a function of the Manev parameter, thus allowing us to extract useful correlations and conclusions. [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]

We have studied the existence, location and stability of the libration points in the model of restricted four-body problem (R4BP) with variable mass. It is assumed that three primaries, one dominant primary and the other two with equal masses, are always forming an equilateral triangle. We have determined the equations of motion of the above mentioned problem for the fourth body which is an infinitesimal mass. The libration points have been determined numerically for different values of the parameters considered. It is found that there are eight or ten libration points out of which six are non-collinear and two or four are collinear depending upon the values of mass parameter and the constant of proportionality occurring in Jeans' law. The regions of motion of the infinitesimal mass have been drawn and investigated. We have also examined the stability of each libration point and found that all the libration points are unstable. Further, the Newton-Raphson basins of attraction are drawn for different set of parameters used. [ABSTRACT FROM AUTHOR]

The present paper deals with the regions of motion and the out-of-plane equilibrium points in the planar restricted three-body problem when both the primaries are heterogeneous oblate spheroids with three layers of different densities and sources of radiation. We have derived the equations of motion and examined the combine effects of radiation and the effect due to the different layers of primaries on the out-of-plane equilibrium points and regions of motion. It is further observed that the energy constant, the radiation parameter and the effect due to the different layers of primaries have substantial impact on the locations of the out-of-plane equilibrium points and on the regions of motion. Furthermore, we have observed that the out-of-plane equilibrium points are unstable for all mass ratios and other parameters, whereas the infinitesimal mass can move freely around the primaries for smaller values of Jacobi constant. But, when we increase the radiation parameter due to either of the primaries, the forbidden regions increase, where the motion is not possible. Moreover, we have discussed the Newton-Raphson basins of attraction to observe the effect of various perturbation on the topology of basins of convergency on configuration plane. [ABSTRACT FROM AUTHOR]

This paper investigates the motion of two infinitesimal masses on the location and stability of the equilibrium points in Robe’s restricted problem of 2 + 2 bodies with the bigger primary a Roche ellipsoid and the smaller a triaxial body. We suppose the bigger primary of mass m1 to be filled with a homogeneous incompressible fluid of density ρ1. The third and the fourth bodies (of mass m3 and m4 respectively) are small solid spheres of density ρ3 and ρ4 respectively inside the ellipsoid, with the assumption that the mass and the radius of the third and the fourth body are infinitesimal. We assume that m2 is describing a circle around m1. The masses m3 and m4 mutually attract each other, do not influence the motion of m1 and m2 but are influenced by them. We have taken into consideration all the three components of the pressure field in deriving the expression for the buoyancy force viz (i) due to the own gravitational field of the fluid (ii) that originating in the attraction of m2 (iii) that arising from the centrifugal force. In this paper, equilibrium solutions of m3 and m4 and their linear stability are analyzed. [ABSTRACT FROM AUTHOR]

We have investigated the nonlinear stability of the triangular libration point L in the R3BP when the smaller primary is a heterogeneous spheroid with three layers having different densities. We observe that in the nonlinear sense, the triangular libration is stable in the range of linear stability 0< μ< μ , a critical value of mass parameter μ, except for three mass ratios $\mu _{1}^{\prime },\mu _{2}^{\prime }$ , $\mu _{3}^{\prime }$ at which Moser's theorem is not applicable. [ABSTRACT FROM AUTHOR]

The effects of the third-order terms ɛ and η in the effective potential V = 1 2 (A x 2 + B y 2) − ɛ x 2 y − η y 3 , on the topology of Newton–Raphson basins of convergence (NRBoC), linked to the equilibrium points are presented. The evolution of the positions of the libration points as a function of the parameters ɛ and η is determined. The basins of convergence (BoC) in (x , y) plane, are unveiled by using the multivariate version of the Newton–Raphson (N-R) iterative scheme. In an attempt to analyse the effect of third order terms on the topology of the BoC as well as the basin entropy, we have performed a systematic numerical investigation. It is revealed that the third-order terms ɛ and η have significant effects on the topology of the BoC linked to the libration points. • The Hénon–Heiles type system has been studied. • The positions of libration points are illustrated. • The basins of convergence are unveiled. • The basins entropy are presented. [ABSTRACT FROM AUTHOR]

The modified circular restricted three-body problem is numerically investigated by exploring the effect of three-body interaction on the basins of convergence connected to the in-plane as well as out-of-plane equilibrium points. The evolution of the positions of the libration points and their stability are illustrated as a function of parameter k due to three-body interaction. It is observed that the number of equilibrium points strongly depends on the sign and the magnitude of the three-body interaction parameter and there exist at most seven libration points where four are non-collinear and three are collinear with the primaries. Moreover, in the Copenhagen case we have found seven collinear libration points. Moreover, the non-collinear as well as collinear libration points are stable for various combinations of mass parameter μ and k. The collinear as well as non-collinear libration points are stable for even those values of μ which are higher than the μ crit of the classical restricted three-body problem. The study of the regions of possible motion shows that as the value of the Jacobian constant decreases, the forbidden region decreases significantly. The attracting domains of the basins of convergence, on several types of two-dimensional planes, are unveiled by applying the multivariate Newton-Raphson iterative scheme. In an attempt to analyse the effect of parameter k on the topology of basins of convergence, a systematic and thorough investigations are presented. The degree of fractality is also unveiled by determining the basin entropy of the convergence plane. • The restricted three-body problem with additional effect of three-body interaction is discussed. • The existence and stability of libration points are discussed. • We numerically investigated the topology of BoCs. • The basin entropy S_b as function of three-body interaction parameter are unveiled. [ABSTRACT FROM AUTHOR]

We have investigated the non-linear stability of the triangular libration point L4 of the Restricted three body problem under the presence of the third and fourth order resonances, when the bigger primary is an oblate body and the smaller a triaxial body and both are source of radiation. It is found through Markeev's theorem that L4 is always unstable in the third order resonance case and stable or unstable in the fourth order resonance case depending upon the values of the parameters A1, A′1, A′2, P and P′, where A1, A′1 and A′2, depends upon the lengths of the semi axes of the primaries and P and P′ are the radiation parameters. [ABSTRACT FROM AUTHOR]

• The regular polygon problem of (N + 1) -bodies with spheroidal primaries are discussed. • Existence and stability of the libration points are discussed. • We unveil the Newton-Raphson basins of convergence, linked to the libration points. • The parametric evolution of the basin entropy is presented. In the present manuscript, we unveil the topology of the basins of convergence in the regular polygon problem of (N + 1) -bodies in two different cases, i.e., in case-I, only the central primary creates the Manev-type quasi-homogeneous potential, and in case-II, the peripheral primaries create the Manev-type quasi-homogeneous potential. The regular polygon problem of (N + 1) -bodies describes the motion of the test particle moving in the force field of N primaries, the ν = N − 1 peripheral primaries of equal masses situated at the vertices of the imaginary regular ν -gon and the Nth primary with different mass i.e., the central primary situated at the centre of mass of the system. In this model, we assume that the primaries create quasi-homogeneous potentials instead of Newtonian potentials and forces. In order to approximate various phenomena due to the irregular shape of the primaries or due to emitting radiation, an inverse cube corrective term is inserted to the inverse square Newtonian law of gravitation. We, numerically, investigated the evolution of the positions of the libration points and their linear stability for different values of ν in both the cases. Further, the multivariate version of Newton-Raphson iterative scheme is applied to unveil the topology of the basins of convergence. Moreover, the "basin entropy" is also computed to analyse the basins of convergence quantitatively. [ABSTRACT FROM AUTHOR]

• The modified R3BP with variable mass are discussed. • Existence and stability of the libration points are discussed. • The parametric evolution of the ZVCs as the function of parameter used is presented. • We unveil the Newton-Raphson basins of convergence, linked to the libration points. In the present work, we shall show an exhaustive numerical analysis of the dynamical behavior of the test particle in the modified restricted three-body problem with variable mass in which the extra effect of the three-body interaction is also taken into account. This additional force ingredient appears in the potential of the classical problem as a new extra term. As the main results, we determine the motion for test particle, under the effect of three-body interaction, which varies its mass according to Jeans' law (Jeans (1928)). The number and existence of the libration points along with their stability have been investigated as the function of value of the parameters which occur due to variable mass of the test particle. Moreover, the regions of the possible motion are also unveiled where the test particle is free to move. Furthermore, the multivariate version of the Newton-Raphson (NR) iterative scheme is used to determine the outcomes of the used parameters on the topology of the basins of convergence (BoC) linked to the libration points. The numerical analysis shows that the topology of the basins of convergence linked with the libration points is highly influenced by the used parameters. Moreover, we perform a systematic analysis to unveil how the regions of convergence are related with the number of required iterations and also with the corresponding probability distributions. [ABSTRACT FROM AUTHOR]

The analysis of the affect of angular velocity on the geometry of the basins of convergence (BoC) linked to the equilibrium points in the restricted three-body problem is illustrated when the primaries are source of radiation. The bivariate scheme of the Newton–Raphson (N–R) iterative method has been used to discuss the topology of the basins of convergence. The parametric evolution of the fractality of the convergence plane is also presented where the degree of fractality is illustrated by evaluating the basin entropy of the convergence plane. • The perturbed restricted three-body problem has been studied. • Existence of the libration points are discussed, as a function of the angular velocity. • We unveil the Newton–Raphson basins of convergence, linked with the libration points. • Basin entropy as function of angular velocity and radiation parameter are illustrated. [ABSTRACT FROM AUTHOR]

In the framework of photogravitational version of the restricted five-body problem, the existence and stability of the in-plane equilibrium points, the possible regions for motion are explored and analysed numerically, under the combined effect of small perturbations in the Coriolis and centrifugal forces. Moreover, the multivariate version of the Newton-Raphson iterative scheme is applied in an attempt to unveil the topology of the basins of convergence linked with the libration points as function of radiation parameters, and the parameters corresponding to Coriolis and centrifugal forces. [ABSTRACT FROM AUTHOR]

In the framework of the axisymmetric problem of restricted five bodies, the existence and stability of the libration points, the regions of possible motion are illustrated and analyzed numerically, under the effect of small perturbations in the Coriolis and centrifugal forces. It is explored how the parameters influence substantially the positions of the libration points and the possible regions of motion. In an attempt to understand how the parameters involved due to the small perturbations in the Coriolis and centrifugal forces affect the stability of the libration points, we perform a systematic investigation and reveal that some of the collinear and non-collinear libration points are stable under these perturbations, whereas none of these libration points are stable for any combination of the angle parameters when the effects of these forces are neglected. [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]

Highlights • The restricted five-body problem with variable mass has been studied. • The locations of the libration points and their stability have been discussed. • The basins of convergence linked with the libration points are discussed. • The Newton–Raphson iterative scheme is used. Abstract In the framework of restricted five-body problem, the existence and stability of the libration points are explored and analysed numerically, under the effect of non–isotropic mass variation of the fifth body (test particle or infinitesimal body). The evolution of the positions of these points are illustrated, as a function of the perturbation parameter. We perform a systematic investigation in an attempt to understand how the perturbation parameter due to variable mass of the fifth body affects the positions, movement and stability of the libration points. In addition, we have revealed how the basins domain of convergence associated with the libration points are substantially influenced by the perturbation parameter. [ABSTRACT FROM AUTHOR]

Highlights • The autonomous restricted four-body problem with variable mass has been studied. • The locations of libration points and their stability have been discussed. • The effect of parameter as well as Jacobian constant on the regions of possible motion are unveiled. • The basins of convergence associated with the libration points using Newton–Raphson iterative scheme are discussed. Abstract We have numerically investigated the circular autonomous restricted four-body problem where the fourth particle of variable mass is moving under the gravitational influence of three bodies known as primaries. Moreover, these primaries move in circular orbit around their common center of mass in such a way that their configuration remains an equilateral triangle configuration. The effect of the parameter α on the existence as well as on the locations of the libration points are investigated. The parametric variation of the positions of the libration points and zero velocity curves are also revealed when the parameter α (which occurs in Jeans' law) increases. Moreover, the Newton–Raphson basins of convergence corresponding to the libration points are unveiled numerically when the parameter α increases. The obtained results strongly suggest that the study of the evolution of the attracting domains of the proposed dynamical system is worth studying in spite of their complexity. [ABSTRACT FROM AUTHOR]

The modified circular restricted three-body problem is numerically investigated to classify the orbits of the test particle. We perform a numerical analysis on the various two-dimensional plane, i.e., (x , C) -plane, (y , C) -plane, and (x , y) -plane to classify the initial conditions on the these planes and distinguish following four types of orbits: (i) the bounded orbits, (ii) collision with the primary m 1 , (iii) collision with the primary m 2 , and (iv) escaping orbits. The motion of the test particle are evaluated numerically, by illustrating the color-coded diagrams (CCDs), where the starting conditions are linked to the orbit type and numerically evaluated as a function of the Jacobian constant C , the initial value of the x − co-ordinate and the Jacobian constant C or the x , y − co-ordinates, or the y − co-ordinate and Jacobian constant. Moreover, in the mean time we have also noted the associated time for classifications of each of the starting conditions on the various 2D-planes. • The restricted three-body problem with the effect of three-body interaction is discussed. • The classification of orbits are discussed. • The regions of possible motion are unveiled. [ABSTRACT FROM AUTHOR]

This manuscript aims to investigate numerically the effect of parameter λ on the basins of convergence (BoCs) associated with the equilibrium points (EPs) of the restricted rhomboidal five-body problem (RR5BP). Moreover, the parametric variation of EPs and zero velocity curves (ZVCs) are also illustrated. Firstly, we have scanned the entire interval for λ ∈ (1 3 , 3) to evaluate the critical value of λ where the number of EPs changes. It is observed that there exist either eleven, thirteen or fifteen EPs in total, however the stability analysis suggests that none of the EPs are linearly stable. The effect of the parameter λ and Jacobian constant C on the regions of possible motion are also illustrated. A systematic numerical investigation is performed to unveil the fact that how the parameter λ affects the geometry of the BoCs. Moreover, we have recorded the total number of iterations needed for each of the initial condition (IC) to converge a specific attractor and shown how the BoCs are related to these iterations and the associated probability distributions. Our numerical results strongly suggest that the parameter λ is indeed very influential factor in this dynamical system. The evolution of attracting regions in this dynamical system is very complicated yet an issue of paramount importance. • The restricted rhomboidal five-body problem are discussed. • Existence of the libration points and zero velocity curves are discussed. • The Newton–Raphson basins of convergence, linked to the libration points are unveiled. • The parametric evolution of the basin entropy and boundary basin entropy are presented. [ABSTRACT FROM AUTHOR]

In the present study, we numerically examined the basins of convergence (BoCs) by deploying the well known Newton–Raphson (NR) iterative scheme, associated to the libration points (LPs) (indeed, act as attractors), in the restricted rhomboidal six-body problem (RRSBP). The parametric evolution of the positions of LPs as a function of the value of parameter " b " is illustrated. Additionally, the linear stability of these LPs and the regions of possible motion are also studied. The BoCs, on the configuration (x , y) plane are unveiled by using the bivariate version of NR-iterative method. Further, a systematic analysis is performed in an order to unveil how the parameter " b " affects the topology as well as degree of fractality of the BoCs. We have also recorded the required number of iterations along with the associated probability distributions (PDs) to show how they are related to the regions of convergence. It is observed that the topology of the BoCs is directly linked with the shape of rhombus. Moreover, basin entropy and basin boundary entropy are also evaluated to unveil the degree of uncertainty of the BoC diagram. • The restricted rhomboidal six-body problem are discussed. • Existence and stability of the libration points are discussed. • The Newton–Raphson basins of convergence, linked to the libration points are unveiled. • The parametric evolution of the basin entropy and boundary basin entropy are presented. [ABSTRACT FROM AUTHOR]

We numerically investigate the effect of oblateness parameter on the topology of basins of convergence connected with the equilibrium points in the restricted three-body problem when the test particle is an oblate spheroid, and the influence of the gravitational potential from the belt is taken into consideration. Additionally, the primaries are also not spherical in shape, on the contrary, it is oblate or prolate spheroid. The parametric variation of the equilibrium points, their stability, and the regions of possible motion are illustrated as the function of the parameters involved. The domain of convergence, on the several two dimensional planes, are unveiled by applying the bi-variate version of the Newton–Raphson iterative method. In addition, we perform a systematic investigation in an order to show how the used parameters affect the topology as well as the degree of fractality of basins of convergence. Moreover, it is also unveiled that how the region of the convergence is related with the number of the required iterations to achieve the desired accuracy with the corresponding probability distribution. • The modified restricted three-body problem has been studied. • The basins of convergence linked to libration points are illustrated. • The degree of fractality of the basins of convergence are evaluated. • The primaries as well as the test particle are oblate/prolate spheroid. [ABSTRACT FROM AUTHOR]

• The axisymmetric restricted five-body problem within the frame of variable mass has been studied. • The positions of libration points are illustrated. • The basins of convergence are unveiled. • The basin entropy is presented. The present manuscript reveals the existence and stability of the libration points (LPs) in the axisymmetric restricted five-body problem (AR5BP) in which mass of the fifth particle varies with respect to time as per Jeans' law. It has been assumed that the four bodies P i with masses m i , i = 1 , 2 , ... , 4 (when m 3 = m 4 = m ˜) form an axisymmetric convex configuration. The system of differential equations which govern the motion of the test particle (whose mass is variable) moving under the gravitational influence of the four primaries, have been presented, indeed, these equations are different from those of the test particle with constant mass in the AR5BP. We have determined the in-plane and out-of-plane LPs along with their linear stability. Furthermore, it is observed that the existence of these LPs depends not only on the angle parameters α and β but also on the parameters occur due to variation in mass namely γ (0 < γ ≤ 1) and σ (0 ≤ σ ≤ 2.2 , a proportionality constant occurs in Jeans' law). Moreover, we have investigated the evolution of the regions of possible motion as function of variable mass parameters where the fifth body can move freely. The topology of the basins of convergence (BoC) linked with the LPs as function of σ is unveiled by deploying the bivariate version of Newton-Raphson (NR) iterative method, and, in order to measure the uncertainty of the basins, the basin entropy is also evaluated. The co-relations between the domains of basins of convergence (DoBoC) and the required number of iterations to achieve predefined accuracy, and also with the corresponding probability distributions are illustrated. [ABSTRACT FROM AUTHOR]

• The restricted three-body problem has been studied. • The primaries are oblate spheroid. • The periodic orbits are investigated. • The displacements along the tangent and normal to the mobile coordinates are given. The present paper deals with the periodic orbits generated by Lagrangian solutions of the restricted three-body problem when both the primaries are oblate bodies. We have illustrated the periodic orbits for different values of μ, h, σ 1 and σ 2 (h is energy constant, μ is mass ratio of the two primaries, σ 1 and σ 2 are oblateness factors). These orbits have been determined by giving displacements along the tangent and normal to the mobile coordinates as defined by (Karimov and Sokolsky, 1989). We have applied the predictor-corrector algorithm to construct the periodic orbits in an attempt to unveil the effect of oblateness of the primaries by taking the fixed values of parameters μ, h, σ 1 and σ 2. [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]

The axisymmetric restricted five-body problem with the concave configuration has been studied numerically to reveal the basins of convergence, by exploring the Newton–Raphson iterative scheme, corresponding to the coplanar libration points (which act as attractors). In addition, four primaries are set in axisymmetric central configurations introduced by Érdi and Czirják [13] and the motion is governed by mutual gravitational attraction only. The evolution of the positions of libration points is illustrated, as a function of the value of angle parameters. A systematic and rigorous investigation is performed in an effort to unveil how the angle parameters affect the topology of the basins of convergence. In addition, the relation of the domain of basins of convergence with required number of iterations and the corresponding probability distributions are illustrated. • The axisymmetric restricted five-body problem: the concave case has been studied. • The locations of the libration points and their stability have been discussed. • The basins of convergence associated with libration points are discussed by using Newton–Raphson iterative scheme. [ABSTRACT FROM AUTHOR]

Abstract In the present work, the Newton–Raphson basins of convergence, corresponding to the coplanar libration points (which act as numerical attractors), are unveiled in the axisymmetric five-body problem, where convex configuration is considered. In particular, the four primaries are set in axisymmetric central configuration, where the motion is governed only by mutual gravitational attractions. It is observed that the total number libration points are either eleven, thirteen or fifteen for different combinations of the angle parameters. Moreover, the stability analysis revealed that all the libration points are linearly stable for all the studied combination of angle parameters. The multivariate version of the Newton–Raphson iterative scheme is used to reveal the structures of the basins of convergence, associated with the libration points, on various types of two-dimensional configuration planes. In addition, we present how the basins of convergence are related with the corresponding number of required iterations. It is unveiled that in almost every case, the basins of convergence corresponding to the collinear libration point L 2 have infinite extent. Moreover, for some combination of the angle parameters, the other collinear libration points L 1 , 2 also have infinite extent. In addition, it can be observed that the domains of convergence, associated with the collinear libration point L 1 , look like exotic bugs with many legs and antennas whereas the domains of convergence, associated with L 4 , 5 look like butterfly wings for some combinations of angle parameters. Particularly, our numerical investigation suggests that the evolution of the attracting domains in this dynamical system is very complicated, yet a worth studying problem. Highlights • The axisymmetric five body problem: the convex configuration has been studied. • The effect of angle parameters on the positions of libration points are discussed. • The Newton–Raphson iterative scheme is used to discuss the basins of convergence. [ABSTRACT FROM AUTHOR]