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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]

• 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]

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]

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]

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 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]

• 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 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]

The Newton–Raphson basins of convergence, corresponding to the coplanar libration points (which act as attractors), are unveiled in the Copenhagen problem, where instead of the Newtonian potential and forces, a quasi-homogeneous potential created by two primaries is considered. The multivariate version of the Newton–Raphson iterative scheme is used to reveal the attracting domain associated with the libration points on various type of two-dimensional configuration planes. The correlations between the basins of convergence and the corresponding required number of iterations are also presented and discussed in detail. The present numerical analysis reveals that the evolution of the attracting domains in this dynamical system is very complicated, however, it is a worth studying issue. [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]