Search

We present an example of demonstration for the basin boundaries in classical rearrangement scattering with particular emphasis on the breakup channel. Whereas the basin boundaries of the other arrangement channels are given by stable manifolds of periodic orbits in the interaction region, the basin boundary of the breakup channel is given by the stable manifold of a particular subset in the set of final asymptotes. The geometry of this boundary surface is presented in detail. Further, we discuss the transition to chaos at the energetic threshold of the breakup channel and the related basin boundary metamorphosis.

We consider a system in which both the parent star and the Earth-like exoplanet move on circular orbits. Using numerical methods, such as the orbit classification technique, we study all types of trajectories of possible exomoons around the exoplanet. In particular, we scan the phase space around the exoplanet and we distinguish between bounded, collisional, and escaping trajectories, considering both retrograde and prograde types of motion. In the case of bounded regular motion, we also use the grid method and a standard predictor-corrector procedure for revealing the corresponding network of symmetric periodic solutions, while we also compute their linear stability.

We use an analytical gravitational model that describes the local motion of stars near the central region of a barred galaxy. By integrating and classifying large sets of starting conditions of trajectories we manage to determine how the important parameters of the bar, such as its mass, strength, scale length, and angular velocity influence the motion of stars. For the orbit taxonomy, we combine traditional dynamical methods such as the classical Poincare surface of section along with modern chaos indicators such as the Smaller Alignment Index (SALI). Our results of the local galactic model are compared with previous studies using global gravitational models.

We consider a Hamiltonian system associated with the two-well umbilical catastrophe potential D5, while the corresponding potential contains only one free parameter. We determine how this free parameter affects the equilibrium dynamics of the system by computing their coordinates on the configuration plane, along with their linear stability and type. Additionally, we discuss the influence of the same free parameter on the orbital dynamics of the system by performing a systematic and thorough orbit classification that allows us to reveal the bounded or escaping motion of the test particle. Our current research project has been funded by Princess Nourah bint Abdulrahman University Researchers, Saudi Arabia Supporting Project number (PNURSP2022R106), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

A systematic study of families of planar symmetric periodic orbits of the elliptic restricted three-body problem is presented, in exoplanetary systems. We find families of periodic orbits that surround only one of the primaries (Satellite-Type), that are moving around both primaries (Planet-Type), and also moving about the collinear Lagrange points. The linear stability of every periodic orbit is calculated, and the families are interpreted through stability diagrams. We focus on quasi-satellite motions of test particles that are associated with the known family [Formula: see text] that consists of 1:1 resonant retrograde Satellite-Type orbits. Over the last years, quasi-satellite orbits are of special interest due to the many applications in the design of spacecraft missions around moons and asteroids. We find the critical simple (1:1 resonant) periodic orbits of the basic families of the circular problem from which we calculate new families of the elliptic problem. Additionally, families of the elliptic problem which bifurcate from the main family [Formula: see text], for various resonances, are also presented and discussed. Hundreds of critical orbits (bifurcation points), from which families of the elliptic problem of higher multiplicity emerge, are found and the corresponding resonances are identified.

We show that the presence of KAM islands in nonhyperbolic chaotic scattering has deep implications on the unpredictability of open Hamiltonian systems. When the energy of the system increases, the particles escape faster. For this reason, the boundary of the exit basins becomes thinner and less fractal. Hence, we could expect a monotonous decrease in the unpredictability as well as in the fractal dimension. However, within the nonhyperbolic regime, fluctuations in the basin entropy have been uncovered. The reason is that when increasing the energy, both the size and geometry of the KAM islands undergo abrupt changes. These fluctuations do not appear within the hyperbolic regime. Hence, the fluctuations in the basin entropy allow us to ascertain the hyperbolic or nonhyperbolic nature of a system. In this manuscript, we have used continuous and discrete open Hamiltonian systems in order to show the relevant role of the KAM islands on the unpredictability of the exit basins, and the utility of the basin entropy to analyze this kind of systems.

The motivation of this article is to numerically investigate the orbital dynamics of the planar post-Newtonian circular restricted problem of three bodies. By numerically integrating several large sets of initial conditions of orbits we obtain the basins of escape. Additionally, we determine the influence of the transition parameter on the orbital structure of the system, as well as on the families of simple symmetric periodic orbits. The networks and the stability of the symmetric periodic orbits are revealed, while the corresponding critical periodic solutions are also identified. The parametric evolution of the horizontal and the vertical stability of the periodic orbits are also monitored, as a function of the transition parameter.

We examine the escape from the four hill potential by using the method of grid classification, when polar coordinates are used for expressing the initial conditions of the orbits. In particular, we investigate how the energy of the orbits influences several aspects of the escape dynamics, such as the escape period and the chosen channels of escape. Color-coded basin diagrams are deployed for presenting the basins of escape using multiple types of planes with two dimensions. We demonstrate that the value of the energy highly influences the escape mechanism of the orbits, as well as the degree of fractality of the dynamical system, which is numerically estimated by computing both the fractal dimension and the entropy of the basin boundaries., Published in Chaos, Solitons & Fractals journal (CSF)

The axisymmetric 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 \'{E}rdi and Czirj\'{a}k 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., Comment: Published in International Journal of Non-Linear Mechanics (IJNLM)

Continuing the series of papers on a new model for a barred galaxy, we investigate the heteroclinic connections between the two normally hyperbolic invariant manifolds sitting over the two index-1 saddle points of the effective potential. The heteroclinic trajectories and the nearby periodic orbits of similar shape populate the bar region of the galaxy and a neighbourhood of its nucleus. Thereby we see a direct relation between the important structures of the interior region of the galaxy and the projection of the heteroclinic tangle into the position space. As a side result, we obtain a detailed picture of the primary heteroclinic intersection surface in the phase space., Comment: Published in Monthly Notices of the Royal Astronomical Society (MNRAS) journal

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., Published in International Journal of Non-Linear Mechanics (IJNLM). arXiv admin note: text overlap with arXiv:1904.04618 and arXiv:1807.00175

The aim of this work is to numerically investigate the nature of the equilibrium points of the axisymmetric five-body problem. Specifically, we consider two cases regarding the convex or concave configuration of the four primary bodies. The specific configuration of the primaries depends on two angle parameters. Combining numerical methods with systematic and rigorous analysis, we reveal how the angle parameters affect not only the relative positions of the equilibrium points but also their linear stability. Our computations reveal that linearly stable equilibria exist in all possible central configurations of the primaries, thus improving and also correcting the findings of previous similar works.

Virus outbreaks have the potential to be a source of severe sanitarian and economic crisis. We propose a new methodology to study the influence of several parameter combinations on the dynamical behavior of simple epidemiological compartmental models. Using this methodology, we analyze the behavior of a simple vaccination model. We find that for susceptible-infected-recovered (SIR) models with seasonality and natural death rate, a new vaccination can reduce the chaoticity of epidemic trajectories, even with nonvaccinated adults. This strategy has little effect on the first infection wave, but it can stop subsequent waves.

In this paper we study analytically the existence of two families of periodic orbits using the averaging theory of second order, and the finite and infinite equilibria of a generalized Henon-Heiles Hamiltonian system which includes the classical Henon-Heiles Hamiltonian. Moreover we show that this generalized Henon-Heiles Hamiltonian system is not C 1 integrable in the sense of Liouville–Arnol'd, i.e. it has not a second C 1 first integral independent with the Hamiltonian. The techniques that we use for obtaining analytically the periodic orbits and the non C 1 Liouville-Arnol'd integrability, can be applied to Hamiltonian systems with an arbitrary number of degrees of freedom.

The character of motion for the three-dimensional circular restricted three-body problem with oblate primaries is investigated. The orbits of the test particle are classified into four types: non-escaping regular orbits around the primaries, trapped chaotic (or sticky) orbits, escaping orbits that pass over the Lagrange saddle points $L_2$ and $L_3$, and orbits that lead the test particle to collide with one of the primary bodies. We numerically explore the motion of the test particle by presenting color-coded diagrams, where the initial conditions are mapped to the orbit type and studied as a function of the total orbital energy, the initial value of the $z$-coordinate and the oblateness coefficient. The fraction of the collision orbits, measured on the color-coded diagrams, show an algebraic dependence on the oblateness coefficient, which can be derived by simple semi-theoretical arguments., Comment: Published in International Journal of Non-Linear Mechanics (IJNLM)

Our work is aimed to investigate the equilibrium dynamics of the restricted three-body problem with equally massed prolate radiating bodies. The positions along with the linear stability of the coplanar points of equilibrium are determined by using numerical methods. In particular, we conduct a systematic and rigorous analysis for revealing the influence of the prolateness coefficient A and the radiation pressure factor q on the dynamics of the system. Our results indicate that both these two parameters are highly influential on the equilibria of the system.

We elucidate the escape properties of stars moving in the combined gravitational field of a binary system of two interacting dwarf spheroidal galaxies. A galaxy model of three degrees of freedom is adopted for describing the dynamical properties of the Hamiltonian system. All the numerical values of the involved parameters are chosen having in mind the real binary system of the dwarf spheroidal galaxies NGC 147 and NGC 185. We distinguish between bounded (regular, sticky or chaotic) and escaping motion by classifying initial conditions of orbits in several types of two dimensional planes, considering only unbounded motion for several energy levels. We analyze the orbital structure of all types of two dimensional planes of initial conditions by locating the basins of escape and also by measuring the corresponding escape time of the orbits. Furthermore, the properties of the normally hyperbolic invariant manifolds (NHIMs), located in the vicinity of the index-1 saddle points $L_1$, $L_2$, and $L_3$, are also investigated. These manifolds are of great importance, as they control the flow of stars (between the two galaxies and toward the exterior region) over the different saddle points. In addition, bifurcation diagrams of the Lyapunov periodic orbits as well as restrictions of the Poincar\'e map to the NHIMs are presented for revealing the dynamics in the neighbourhood of the saddle points. Comparison between the current outcomes and previous related results is also made., Comment: Published in International Journal of Non-Linear Mechanics (IJNLM)

We explore a pseudo-Newtonian planar circular restricted three-body problem in which the primaries are modeled using an approximate gravitational potential up to the second nonvanishing term of the Fodor–Hoenselaers–Perjés expansion. We aim to understand how the main free parameters of the system affect its dynamical properties. In particular, we determine how the mass of the primaries as well as the transition parameters affect not only the properties of the points of equilibrium (total number, locations, and linear stability) but also the networks of simple symmetric periodic orbits. Our results show that, under this approach, significant variations are observed in the fixed points (number and stability) and periodic orbits of the planar circular restricted three-body problem, even when small contributions of the non-Newtonian terms are considered. We also provide direct applications of the new model potential in real observable binary stellar systems.

In this work we numerically investigate the planar circular restricted three-body problem and apply this model to the Sun-Jupiter-particle problem where the particle is orbiting Jupiter. Our aim is to complement the qualitative mapping technique with quantitative techniques to trace a given trajectory in phase space. Qualitatively, this problem can be investigated using the Poincare surface of sections mapping technique. Here we compute such maps for various Jacobian energies. While the computation of classic Poincare surface of sections are useful to qualitatively classify phase-space regions with periodic, quasi-periodic, or chaotic motion, the method is inadequate to describe escape and/or collision orbits. To mitigate this shortcoming we complement such maps with the calculations of quantitative dynamical maps based on the Smaller-Alignment Index (SALI) technique. This allows for a complete assessment of the global dynamics of the problem resulting in a detailed classification of orbits of differing dynamical character. We revealed the network of simple periodic orbits around Jupiter, along with their linear stability. As a highlight, we identified a region of flipping orbits that are not detected with Poincare surface of sections. We outline and discuss various assumptions. After a short review of the underlying model and applied numerical techniques, we present and discuss results from this work.

The size distribution and orbital architecture of dust, grains, boulders, asteroids, and major planets during the giant branch phases of evolution dictate the preponderance and observability of the eventual debris, which have been found to surround white dwarfs and pollute their atmospheres with metals. Here, we utilize the photogravitational planar restricted three-body problem in one-planet giant branch systems in order to characterize the orbits of grains as the parent star luminosity and mass undergo drastic changes. We perform a detailed dynamical analysis of the character of grain orbits (collisional, escape, or bounded) as a function of location and energy throughout giant branch evolution. We find that for stars with main-sequence masses of $2.0M_{\odot}$, giant branch evolution, combined with the presence of a planet, ubiquitously triggers escape in grains smaller than about 1 mm, while leaving grains larger than about 5 cm bound to the star. This result is applicable for systems with either a terrestrial or giant planet, is largely independent of the location of the planet, and helps establish a radiative size threshold for escape of small particles in giant branch planetary systems., Comment: Published in Astronomy & Astrophysics journal (A&A)

Nearly all known white dwarf planetary systems contain detectable rocky debris in the stellar photosphere. A glaring exception is the young and still evolving white dwarf WD J0914+1914, which instead harbours a giant planet and a disc of pure gas. The stability boundaries of this disc and the future prospects for this white dwarf to be polluted with rocks depend upon the mass and orbit of the planet, which are only weakly constrained. Here we combine an ensemble of plausible planet orbits and masses to determine where observers should currently expect to find the outer boundary of the gas disc. We do so by performing a sweep of the entire plausible phase space with short-term numerical integrations. We also demonstrate that particle-star collisional trajectories, which would lead to the (unseen) signature of rocky metal pollution, occupy only a small fraction of the phase space, mostly limited to particle eccentricities above 0.75. Our analysis reveals that a highly inflated planet on a near-circular orbit is the type of planet which is most consistent with the current observations., Comment: Accepted for publication in Monthly Notices of the Royal Astronomical Society Main Journal

The scope of this work is to perform a numerical investigation of the orbital dynamics for a test particle in the pseudo-Newtonian Hill problem. Large two-dimensional sets of initial conditions of prograde and retrograde orbits are investigated. The orbits are classified as bounded (chaotic, sticky or regular), escaping and collision orbits. The smaller alignment index (SALI) method is used to identify chaotic orbits. Additionally, the influence of the energy (or equivalently the value of the Jacobi constant) and of the Schwarzschild radius on the orbital structure of the system are determined. Our numerical results are compared with related previous ones, corresponding to the classical version of the Hill problem.

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)

The escape mechanism of orbits in a star cluster rotating around its parent galaxy in a circular orbit is investigated. A three degrees of freedom model is used for describing the dynamical properties of the Hamiltonian system. The gravitational field of the star cluster is represented by a smooth and spherically symmetric Plummer potential. We distinguish between ordered and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. The Smaller Alignment Index (SALI) method is used for determining the regular or chaotic nature of the orbits. The basins of escape are located and they are also correlated with the corresponding escape time of the orbits. Areas of bounded regular or chaotic motion and basins of escape were found to coexist in the $(x,z)$ plane. The properties of the normally hyperbolic invariant manifolds (NHIMs), located in the vicinity of the index-1 Lagrange points $L_1$ and $L_2$, are also explored. These manifolds are of paramount importance as they control the flow of stars over the saddle points, while they also trigger the formation of tidal tails observed in star clusters. Bifurcation diagrams of the Lyapunov periodic orbits as well as restrictions of the Poincar\'e map to the NHIMs are deployed for elucidating the dynamics in the neighbourhood of the saddle points. The extended tidal tails, or tidal arms, formed by stars with low velocity which escape through the Lagrange points are monitored. The numerical results of this work are also compared with previous related work., Comment: Published in Monthly Notices of the Royal Astronomical Society (MNRAS) journal

A three degrees of freedom (3-dof) barred galaxy model composed of a spherically symmetric nucleus, a bar, a flat disc and a spherically symmetric dark matter halo is used for investigating the dynamics of the system. We use colour-coded plots to demonstrate how the value of the semi-major axis of the bar influences the regular or chaotic dynamics of the 3-dof system. For distinguishing between ordered and chaotic motion we use the Smaller ALingment Index (SALI) method, a fast yet very accurate tool. Undoubtedly, the most important elements of the dynamics are the normally hyperbolic invariant manifolds (NHIMs) located in the vicinity of the index 1 Lagrange points $L_2$ and $L_3$. These manifolds direct the flow of stars over the saddle points, while they also trigger the formation of rings and spirals. The dynamics in the neighbourhood of the saddle points is visualized by bifurcation diagrams of the Lyapunov orbits as well as by the restriction of the Poincar\'e map to the NHIMs. In addition, we reveal how the semi-major axis of the bar influences the structure of these manifolds which determine the final stellar structure (rings or spirals). Our numerical simulations suggest that in galaxies with weak bars the formation of $R_1$ rings or $R_1'$ pseudo-rings is favoured. In the case of galaxies with intermediate and strong bars the invariant manifolds seem to give rise to $R_1R_2$ rings and twin spiral formations, respectively. We also compare our numerical outcomes with earlier related work and with observational data., Comment: Published in Monthly Notices of the Royal Astronomical Society (MNRAS) journal

The three-dimensional version of the circular restricted problem of three bodies is utilized to describe a system comprising a host star and an exoplanet. The third body, playing the role of a test particle, can be a comet or an asteroid, or even a small exomoon. Combining the grid classification method with two-dimensional color-coded basin maps, we determine the nature of the motion of the test particle by distinguishing between collision, escaping, and bounded motion. In the case of ordered bounded motion, we also obtain the orientation (retrograde or prograde) as well as the geometry (circulating around one or both of the two main bodies) of the trajectories of the third body, which starts from either the pericenter or apocenter. Following this approach, we are able to systematically explore the dependence of the motion type of the test particle on the initial values of the semimajor axis, eccentricity, and inclination of its orbit.

The equilateral restricted four-body problem, with triaxial rigid oblate bodies, is investigated. Using numerical methods we examine how the linear stability and the positions of the coplanar libration points are affected by the triaxility parameters of the primaries. In each case, we perform a rigorous and systematic analysis for determining the influence of the triaxility parameters σ 1 and σ 2 on the dynamics of the system. Our computations suggest the strong influence of these parameters by revealing additional cases, regarding the total number of equilibria, thus improving the findings of previous related works.

The on-plane version of the restricted problem of 3 bodies with a continuation fraction potential is numerically investigated. The idea is to consider that one of the primaries is a radiation source and the secondary one is not spherical. By adopting the grid classification method we locate the coordinates, on the X Y − plane, of all the points of equilibrium, for several values of the involved parameters. The stability of the libration points is also computed, as a function of the same parameters. The shape as well as the properties of the Newton–Raphson basins of convergence, associated with the equilibria of the system, are also explored for obtaining the optimal starting conditions of the iterator. Our analysis reveals that the new potential has additional points of equilibrium, with respect to the classical 3-body problem.

We reveal the networks of simple symmetric periodic orbits in a double-barred galaxy model. Specifically, we investigate the dependence on the total orbital energy of the positions but also on the stability of the periodic solutions. For every orbital family, we also compute the horizontal and vertical critical parameter values of the system, at which new periodic families bifurcate from. Of particular interest are the vertical critical points which act as starting points for the creation of new families of three-dimensional periodic orbits. The atlas of the simple periodic trajectories is presented in the (x, E) plane and also in the (x, E, z) and ( x , E , z ˙ ) spaces, in order to obtain the global parametric evolution of the orbital families. Our analysis suggests that the orbital properties of a double-barred galaxy model are very complicated and at the same time very fascinating.

The planar version of the equilateral restricted four-body problem, with three unequal masses, is numerically investigated. By adopting the grid classification method we locate the coordinates, on the plane [Formula: see text], of the points of equilibrium, for all possible values of the masses of the primaries. The linear stability of the libration points is also determined, as a function of the masses. Our analysis indicates that linearly stable points of equilibrium exist only when one of the primaries has a considerably larger mass, with respect to the other two primary bodies, when the triangular configuration of the primaries is also dynamically stable.

The dynamics of the orbital motion in the planar elliptic restricted three-body problem are investigated, by using the method of grid classification. In this system, the secondary body is an exoplanet, while the corresponding primary body is its parent star. We numerically investigate how several dynamical quantities of the system, such as the orbital energy, the eccentricity, the true anomaly, and the mass parameter, influence several aspects of the motion of the test particle, such as the final state as well as the time of escape/collision of the orbits. Color-coded basin diagrams are utilized for displaying all the different types of basins, using two-dimensional maps. The results of this analysis are then compared to similar ones from the classical version of the circular problem of three bodies.

This article aims to investigate the points of equilibrium and the associated convergence basins in the restricted problem with two primaries, with a modified gravitational potential. In particular, for one of the primary bodies, we add an external gravitational term of the form 1/r3, which is very common in general relativity and represents a gravitational field much stronger than the classical Newtonian one. Using the well-known Newton–Raphson iterator we numerically locate the position of the points of equilibrium, while we also obtain their linear stability. Furthermore, for the location of the points of equilibrium, we obtain semi-analytical functions of both the mass parameter and the transition parameter. Finally, we demonstrate how these two variable parameters affect the convergence dynamics of the system as well as the fractal degree of the basin diagrams. The fractal degree is derived by computing the (boundary) basin entropy.

We examine the orbital dynamics of the planar pseudo-Newtonian Copenhagen problem, in the case of a binary system of Schwarzschild-like primaries, such as super-massive black holes. In particular, we investigate how the Jacobi constant (which is directly connected with the energy of the orbits) influences several aspects of the orbital dynamics, such as the final state of the orbits. We also determine how the relativistic effects (i.e., the Schwarzschild radius) affect the character of the orbits, by comparing our results with the classical Newtonian problem. Basin diagrams are deployed for presenting all the different basin types, using multiple types of planes with two dimensions. We demonstrate that both the Jacobi constant as well as the Schwarzschild radius highly influence the character of the orbits, as well as the degree of fractality of the dynamical system., Comment: Published in Monthly Notices of the Royal Astronomical Society (MNRAS) journal

We consider the planar Hill's lunar problem with a homogeneous gravitational potential. The investigation of the system is twofold. First, the starting conditions of the trajectories are classified into three classes, that is bounded, escaping, and collisional. Second, we study the no-return property of the Lagrange point $L_2$ and we observe that the escaping trajectories are scattered exponentially. Moreover, it is seen that in the supercritical case, with $\alpha \geq 2$, the basin boundaries are smooth. On the other hand, in the subcritical case, with $\alpha < 2$ the boundaries between the different types of basins exhibit fractal properties., Comment: 12 pages, 10 figures

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:1806.11414

The circular Sitnikov problem, where the two primary bodies are prolate or oblate spheroids, is numerically investigated. In particular, the basins of convergence on the complex plane are revealed by using a large collection of numerical methods of several order. We consider four cases, regarding the value of the oblateness coefficient which determines the nature of the roots (attractors) of the system. For all cases we use the iterative schemes for performing a thorough and systematic classification of the nodes on the complex plane. The distribution of the iterations as well as the probability and their correlations with the corresponding basins of convergence are also discussed. Our numerical computations indicate that most of the iterative schemes provide relatively similar convergence structures on the complex plane. However, there are some numerical methods for which the corresponding basins of attraction are extremely complicated with highly fractal basin boundaries. Moreover, it is proved that the efficiency strongly varies between the numerical methods., Comment: Published in Astrophysics and Space Science (A&SS) journal