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

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

Basins of convergence of equilibrium points in the pseudo-Newtonian planar circular restricted three-body problem

The Newton-Raphson basins of attraction, associated with the libration points (attractors), are revealed in the pseudo-Newtonian planar circular restricted three-body problem, where the primaries have equal masses. The parametric variation of the position as well as of the stability of the equilibrium points is determined, when the value of the transition parameter $\epsilon$ varies in the interval $[0,1]$. The multivariate Newton-Raphson iterative scheme is used to determine the attracting domains on several types of two-dimensional planes. A systematic and thorough numerical investigation is performed in order to demonstrate the influence of the transition parameter on the geometry of the basins of convergence. The correlations between the basins of attraction and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly indicates that the evolution of the attracting regions in this dynamical system is an extremely complicated yet very interesting issue.Comment: Published in Astrophysics and Space Science (A&SS) journal. Previous papers with related context: arXiv:1702.07279, arXiv:1706.07044, arXiv:1704.02273, arXiv:1709.0663

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

On the Newton-Raphson basins of convergence of the out-of-plane equilibrium points in the Copenhagen problem with oblate primaries

The Copenhagen case of the circular restricted three-body problem with oblate primary bodies is numerically investigated by exploring the Newton-Raphson basins of convergence, related to the out-of-plane equilibrium points. The evolution of the position of the libration points 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 multivariate Newton-Raphson iterative method. We perform a systematic and thorough investigation in an attempt to understand how the oblateness coefficient affects the geometry of the basins of convergence. The convergence regions are also related with the required number of iterations and also with the corresponding probability distributions. The degree of the fractality is also determined by calculating the fractal dimension and the basin entropy of the convergence planes.Comment: Published in International Journal of Non-Linear Mechanics (IJNLM). arXiv admin note: text overlap with arXiv:1801.01378, arXiv:1806.11409, arXiv:1807.0069

Equilibrium points and basins of convergence in the linear restricted four-body problem with angular velocity

The planar linear restricted four-body problem is used in order to determine the Newton-Raphson basins of convergence associated with the equilibrium points. The parametric variation of the position as well as of the stability of the libration points is monitored when the values of the mass parameter $b$ as well as of the angular velocity $\omega$ vary in predefined intervals. The regions on the configuration $(x,y)$ plane occupied by the basins of attraction are revealed using the multivariate version of the Newton-Raphson iterative scheme. The correlations between the attracting domains of the equilibrium points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We perform a thorough and systematic numerical investigation by demonstrating how the parameters $b$ and $\omega$ influence the shape, the geometry and of course the fractality of the converging regions. Our numerical outcomes strongly indicate that these two parameters are indeed two of the most influential factors in this dynamical system.Comment: Published in Chaos, Solitons and Fractals journal (CSF). arXiv admin note: previous papers with related context: arXiv:1702.07279, arXiv:1704.0227

Orbit classification and networks of periodic orbits in the planar circular restricted five-body problem

The aim of this paper is to numerically investigate the orbital dynamics of the circular planar restricted problem of five bodies. By numerically integrating several large sets of initial conditions of orbits we classify them into three main categories: (i) bounded (regular or chaotic) (ii) escaping and (iii) close encounter orbits. In addition, we determine the influence of the mass parameter on the orbital structure of the system, on the degree of fractality, as well as on the families of symmetric and non-symmetric periodic orbits. The networks and the linear stability of both symmetric and non-symmetric periodic orbits are revealed, while the corresponding critical periodic solutions are also identified. The parametric evolution of the horizontal and the vertical linear stability of the periodic orbits is also monitored, as a function of the mass parameter.Comment: Published in International Journal of Non-Linear Mechanics (IJNLM

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

Positions of equilibrium points for dust particles in the circular restricted three-body problem with radiation

For a body with negligible mass moving in the gravitational field of a star and one planet in a circular orbit (the circular restricted three-body problem) five equilibrium points exist and are known as the Lagrangian points. The positions of the Lagrangian points are not valid for dust particles because in the derivation of the Lagrangian points is assumed that no other forces beside the gravitation act on the body with negligible mass. Here we determined positions of the equilibrium points for the dust particles in the circular restricted three-body problem with radiation. The equilibrium points are located on curves connecting the Lagrangian points in the circular restricted three-body problem. The equilibrium points for Jupiter are distributed in large interval of heliocentric distances due to its large mass. The equilibrium points for the Earth explain a cloud of dust particles trailing the Earth observed with the Spitzer Space Telescope. The dust particles moving in the equilibrium points are distributed in interplanetary space according to their properties.Comment: 19 pages, 7 figures, accepted for publication in MNRA

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