The perturbed restricted three-body problem with angular velocity: Analysis of basins of convergence linked to the libration points

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

On the perturbed photogravitational restricted five-body problem: the analysis of fractal basins of convergence

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.Comment: 12 Figur

The analysis of restricted five-body problem within frame of variable mass

In the framework of restricted five bodies 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 and the possible regions of motion 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 domain of the basins of convergence associated with the libration points are substantially influenced by the perturbation parameter

Revealing the Newton-Raphson basins of convergence in the circular pseudo-Newtonian Sitnikov problem

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.Comment: Published in International Journal of Non-Linear Mechanics (IJNLM). arXiv admin note: text overlap with arXiv:1807.00693, arXiv:1806.1140

Unveiling the basins of convergence in the pseudo-Newtonian planar circular restricted four-body problem

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 $\epsilon \in [0, 1]$. The stability of these libration points has also been discussed. Our study reveals that the Jacobi constant as well as transition parameter $\epsilon$ 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.Comment: Published in New Astronomy journa

Comparing the geometry of the basins of attraction, the speed and the efficiency of several numerical methods

We use simple equations in order to compare the basins of attraction on the complex plane, corresponding to a large collection of numerical methods, of several order. Two cases are considered, regarding the total number of the roots, which act as numerical attractors. For both cases we use the iterative schemes for performing a thorough and systematic classification of the nodes on the complex plane. The distributions of the required iterations as well as the probability and their correlations with the corresponding basins of convergence are also discussed. Our numerical calculations suggest that most of the iterative schemes provide relatively similar convergence structures on the complex plane. In addition, several aspects of the numerical methods are compared in an attempt to obtain general conclusions regarding their speed and efficiency. Moreover, we try to determine how the complexity of the each case influences the main characteristics of the numerical methods.Comment: Published in International Journal of Applied and Computational Mathematics (IACM). arXiv admin note: text overlap with arXiv:1806.1141

On the convergence dynamics of the Sitnikov problem with non-spherical primaries

We investigate, using numerical methods, the convergence dynamics of the Sitnikov problem with non-spherical primaries, by applying the Newton-Raphson (NR) iterative scheme. In particular, we examine how the oblateness parameter $A$ influences several aspects of the method, such as its speed and efficiency. Color-coded diagrams are used for revealing the convergence basins on the plane of complex numbers. Moreover, we compute the degree of fractality of the convergence basins on the complex space, as a relation of the oblateness, by using different computational tools, such the fractal dimension as well as the (boundary) basin entropy.Comment: Published in International Journal of Applied and Computational Mathematics (IACM

Investigating the basins of convergence in the circular Sitnikov three-body problem with non-spherical primaries

In this work we numerically explore the Newton-Raphson basins of convergence, related to the equilibrium points, in the Sitnikov three-body problem with non-spherical primaries. The evolution of the position of the roots is determined, as a function of the value of the oblateness coefficient. The attracting regions, on several types of two dimensional planes, are revealed by using the classical Newton-Raphson iterative method. We perform a systematic and thorough investigation in an attempt to understand how the oblateness coefficient affects the geometry as well as the overall properties of the convergence regions. The basins of convergence are also related with the required number of iterations and also with the corresponding probability distributions.Comment: Published in Few-Body Systems (FBSY) journal. arXiv admin note: substantial text overlap with arXiv:1806.11409; text overlap with arXiv:1801.01378, arXiv:1801.00710, arXiv:1803.07398, arXiv:1702.0727

Basins of convergence in the circular Sitnikov four-body problem with non-spherical primaries

The Newton-Raphson basins of convergence, related to the equilibrium points, in the Sitnikov four-body problem with non-spherical primaries are numerically investigated. We monitor the parametric evolution of the positions of the roots, as a function of the oblateness coefficient. The classical 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 oblateness coefficient 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.Comment: Published in International Journal of Bifurcation and Chaos (IJBC) journa

On the fractal basins of convergence of the libration points in the axisymmetric five-body problem: the convex configuration

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 combination of the angle parameters. Moreover, the stability analysis revealed that the 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 cases, the basins of convergence corresponding to the collinear libration point $L_2$ have infinite extent. Moreover, for some combination of the angle parameters, the collinear libration points $L_{1,2}$ have also 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 worthy studying problem.Comment: Published in International Journal of Non-Linear Mechanics (IJNLM). arXiv admin note: text overlap with arXiv:1904.04618 and arXiv:1807.0017