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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]

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 has been studied. • The parametric evolution of the stationary points are illustrated. • The effect of the Coriolis and centrifugal force on the stationary points and their stability are unveiled. • The multivariate version of the NR-iterative scheme is used to discuss the basins of convergence. In the present manuscript, the effects of small perturbations in the centrifugal and Coriolis forces with additional effect of three-body interaction on the locations as well as stability of the stationary points (SPs), and on the topology of the basins of convergence (BoCs) linked with these SPs in the restricted three-body problem (R3BP) are discussed by using the multivariate version of the Newton-Raphson iterative scheme. It is found that the positions of the SPs are highly influenced by the centrifugal force but their stability is highly influenced by combinations of the centrifugal and Coriolis forces. In addition, the correlations betwixt the basin of convergence (BoC) connected to the SPs and the associated number of iterations needed to assure pre-assumed precision are analyzed. Our analysis suggests that the mass ratio μ , three-body interaction parameter k and the effect in centrifugal and the Coriolis forces controlled by parameters ϕ and φ are very influential parameters. [ABSTRACT FROM AUTHOR]

This research article presents the effect of the Earth's equatorial ellipticity and the resistive force on a geocentric satellite's resonant curves. The resonance occurs in the motion of the satellite due to the frequencies, the angular velocity of the satellite, the rate of change of Earth's equatorial ellipticity parameter (γ ̇) , the steady-state angular velocity of the Moon around the Earth ( ϑ 0 ̇) , and the steady-state angular velocity of barycenter (Θ ̇) around the Sun. We express the equations of motion of the satellite in a spherical coordinate system by using the Earth's potential. By applying the unperturbed solution, we reduced the system of equations into a second-order differential equation and obtained its solution. From the solution it is observed that the resonance appears between the frequencies δ s 0 ̇ , γ ̇ , Θ 0 ̇ and ϑ 0 ̇ at ten resonant points. We analyze the resonant curves corresponding to the resonant points and show the effect of resistive force and latitude of a satellite. Also, We show the effect of eccentricity (e) and semi-major axis (a) on the oscillatory amplitudes (A i) , for i = 1 , 2 , 3 ... 21. [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]

• The position of equilibrium points are monitored with respect to the parameters the distances between galaxies (R), the masses of disk and nucleus of galaxies (M d 1 , M n 1 , M d 2 , M n 2 ). • The zero velocity curves are investigated under the effect of above parameters. • We determine the Newton-Raphson's basins of attraction associated with the libration points. • The existence of unpredictable (fractal) regions are explored using the method of basin entropy. • Due to variation of the parameter R, the change in basin entropy (S b) and boundary basin entropy (S bb) is graphically shown. The present paper investigates the binary system of quasars in the framework of the Circular Restricted Three-Body Problem. The parametric evolution of libration points and the geometry of zero-velocity curves are one of the crucial aspects of our study. The multivariate form of NR method is applied to study the basin of attraction connected with libration points. The algorithm for using the Newton-Raphson method is slightly modified in order to avoid the unnecessary delay in the convergence of initial conditions. The impact of parameters on the pattern of the basin of attraction and the number of iterations needed for the convergence of initial conditions are explored. We carry out an exhaustive (numerical) study to show the influence of these parameters on converging regions in basins of attraction. We unveil the existence of fractal structure in the basin of attraction using the method of basin entropy. In almost all cases, the existence of fractal structure is found throughout the basins of attraction. [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 this paper, a HIV/AIDS and TB co-infection model has been explored, which incorporates detection and treatment for both diseases. We begin with presenting a co-infection model and then start analyzing both the HIV and the TB sub-models separately. The basic reproduction numbers corresponding to both HIV and TB are computed. Both the HIV/AIDS and the TB sub-models have been shown to exhibit two equilibrium points, namely, the disease-free equilibrium point and the unique endemic equilibrium point. For both sub-models, the disease-free equilibrium point is locally as well as globally asymptotically stable when their corresponding reproduction number is less than unity. The endemic equilibrium point for both sub-models exists, when their corresponding reproduction numbers is greater than one. We also analyze the full HIV-TB co-infection model. With the aim of minimizing infectives and the cost of applying effort towards the detection and the treatment, optimal control analysis is performed for the full model using the Pontryagin's maximum principle. Numerical simulations with different combinations of efforts are then performed to explore the effect of detection in the presence of treatment for both diseases. Numerical simulations emphasize the fact that to reduce co-infection from the population, programs to accelerate the detection of both diseases are also required along with the treatment. • This paper presents a non-linear HIV-TB co-infection model. • Local stability analysis for the model has been done. • An optimal control problem to minimize the total cost is presented. • The existence and characterization of the optimal controls has been established. [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]

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]

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]

In this paper we numerically explore the convergence properties of the pseudo-Newtonian circular restricted problem of three and four primary bodies. The classical 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 transition parameter on the convergence properties of the system. Additionally, the roots (numerical attractors) of the system and the basin entropy of the convergence diagrams are monitored as a function of the transition parameter, thus allowing us to extract useful conclusions. The probability distributions, as well as the distributions of the required number of iterations are also correlated with the corresponding 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]