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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]

We have studied the restricted four-body problem (R4BP) with the effect of the small perturbation in the Coriolis and centrifugal forces on the libration points and zero velocity curves (ZVCs). Further, we have supposed that all the primaries are set in an equilateral triangle configuration, moving in the circular orbits around their common centre of mass. We have observed that the effect of the small perturbation in centrifugal force has a substantial effect on the location of libration points but a small perturbation in the Coriolis force has no impact on the location of libration points. But the stability of the libration points is highly influenced by the effect of the small perturbation in the Coriolis force. It is observed that as the Coriolis parameter increases, the libration points become stable. Further, it is found that the effect of the small perturbation in the centrifugal force has a substantial influence on the regions of possible motion. Also, when the effect of small perturbation in the centrifugal force increases the forbidden region decreases; here the motion is not possible for the infinitesimal mass. It is observed when the value of the Jacobian constant decreases, the regions of possible motion increase. In addition, we have also discussed how small perturbations in the Coriolis and centrifugal forces influence the Newton-Raphson basins of convergence. [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]

• 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]

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]

This paper deals with the photogravitational circular restricted three-body problem, including the effect of circular cluster of material points. The parametric evolution of the position of equilibrium points and the existence of a different number of equilibrium points are illustrated using graphs. A detailed investigation to find out the influence of the radiation pressure q , mass ratio μ and the circular cluster of material points M b on the geometry of basins of attraction is carried out. Further, the number of iterations required to obtain the desired level of accuracy is recorded and presented using probability distribution diagram. The correlations between the domain of convergence of equilibrium points and the corresponding number of the iterations required to obtain the desired level of accuracy are explained. We monitor the parametric evolution of the basin entropy to reveal the unpredictability in the basins of attractions. It is found that unpredictable (fractal) regions exist in the vicinity of boundaries of the basins of attraction. [ABSTRACT FROM AUTHOR]

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

In 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 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]

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]

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]