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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

We numerically investigate the motion of stars on the meridional plane of an axially symmetric disk galaxy model, containing a central supermassive black hole, represented by the Paczynski-Wiita potential. By using this pseudo-Newtonian potential we can replicate important relativistic properties, such as the existence of the Schwarzschild radius. After classifying extensive samples of initial conditions of trajectories, we manage to distinguish between collisional, ordered, and chaotic motion. Besides, all starting conditions of regular orbits are further classified into families of regular orbits. Our results are presented through modern color-coded basin diagrams on several types of two-dimensional planes. Our analysis reveals that both the mass of the black hole (in direct relation with the Schwarzschild radius) as well as the angular momentum play an important role in the character of orbits of stars. More specifically, the trajectories of low angular momentum stars are highly affected by the mass of the black hole, while high angular momentum stars seem to be unaffected by the central black hole. Comparison with previous related outcomes, using Newtonian potentials for the central region of the galaxy, is also made., 10 pages, 10 figures

In this paper, we consider a Newtonian system whose relativistic counterpart describes a superimposed halo with a black hole. Our aim is to determine how the quadrupole and octupole moments affect the nature of the motion of a test particle, moving in the close vicinity of the black hole. The different types of trajectories for the test particle are mainly classified as bounded, collisional, and escaping, by using modern color-coded basin diagrams. Moreover, an additional analysis is carried out for distinguishing between the different types of bounded motion (regular, sticky, and chaotic). Our results strongly indicate that the multipole moments, along with the total orbital energy, highly affect the final state of the test particle, while at the same time the basin geometry of the phase space tends to be highly dominated by collision and escape orbits., 10 pages, 7 figures

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)

This paper deals with the derivation and analysis of a seventh-order generalization of the H\'enon-Heiles potential. The new potential has axial and reflection symmetries, and finite escape energy with three channels of escape. Based on SALI indicator and exits basins, the dynamic behavior of the seventh-order system is investigated qualitatively in cases of bounded and unbounded movement. Moreover, a quantitative analysis is carried out through the percentage of chaotic orbits and the basin entropy, respectively. After classifying large sets of initial conditions of orbits for several values of the energy constant in both regimes, we observe that when the energy moves away from the critical value, the chaoticity of the system decreases and the basin structure becomes simpler with sharper and well defined bounds. Our results suggest that when the seventh-order contributions of the potential are taken into account, the system becomes less ergodic in comparison with the classical version of the H\'enon-Heiles system., Comment: 10 pages, 7 figures

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

We examine the orbital dynamics in a new Hénon–Heiles system with an additional gravitational potential, by classifying sets of starting conditions of trajectories. Specifically, we obtain the results on how the total orbital energy along with the transition parameter influence the overall dynamics of the massless test particle, as well as the respective time of escape/collision. By using modern diagrams with color codes we manage to present the different types of basins of the system. We show that the character of the orbits is highly dependent on the energy and the transition parameter. This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia, under the grant number KEP-17-130-41. The authors, therefore, gratefully acknowledge DSR for technical and financial support.

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.

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 journal

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.

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 dynamics of a test particle in a non-spinning binary black hole system of equal masses is numerically investigated. The binary system is modeled in the context of the pseudo-Newtonian circular restricted three-body problem, such that the primaries are separated by a fixed distance and move in a circular orbit around each other. In particular, the Paczy\'{n}ski-Wiita potential is used for describing the gravitational field of the two non-Newtonian primaries. The orbital properties of the test particle are determined through the classification of the initial conditions of the orbits, using several values of the Jacobi constant, in the Hill's regions of possible motion. The initial conditions are classified into three main categories: (i) bounded, (ii) escaping and (iii) displaying close encounters. Using the smaller alignment index (SALI) chaos indicator, we further classify bounded orbits into regular, sticky or chaotic. To gain a complete view of the dynamics of the system, we define grids of initial conditions on different types of two-dimensional planes. The orbital structure of the configuration plane, along with the corresponding distributions of the escape and collision/close encounter times, allow us to observe the transition from the classical Newtonian to the pseudo-Newtonian regime. Our numerical results reveal a strong dependence of the properties of the considered basins with the Jacobi constant as well as with the Schwarzschild radius of the black holes., Comment: Published in Monthly Notices of the Royal Astronomical Society (MNRAS) journal

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., Published in International Journal of Non-Linear Mechanics (IJNLM)

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., Published in International Journal of Non-Linear Mechanics (IJNLM). arXiv admin note: text overlap with arXiv:1801.01378, arXiv:1806.11409, arXiv:1807.00693

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

The planar circular restricted three-body problem with modified Newtonian gravity is used in order to determine the Newton-Raphson basins of attraction associated with the equilibrium points. The evolution of the position of the five Lagrange points is monitored when the value of the power $p$ of the gravitational potential of the second primary varies 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 basins of convergence of the equilibrium points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We conduct a thorough and systematic numerical investigation by demonstrating how the dynamical quantity $p$ influences the shape as well as the geometry of the basins of attractions. Our results strongly suggest that the power $p$ is indeed a very influential parameter in both cases of weaker or stronger Newtonian gravity., Published in Journal of Applied Mathematics and Computing (JAMC). Previous papers with related context: arXiv:1608.08610, arXiv:1704.02273, arXiv:1702.07279, arXiv:1706.07044

We investigate in detail the escape dynamics in an analytical gravitational model which describes the motion of stars in a quasar galaxy with a disk and a massive nucleus. We conduct a thorough numerical analysis distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. In order to distinguish safely and with certainty between ordered and chaotic motion we apply the Smaller ALingment Index (SALI) method. It is of particular interest to locate the escape basins through the openings around the collinear Lagrangian points $L_1$ and $L_2$ and relate them with the corresponding spatial distribution of the escape times of the orbits. Our exploration takes place both in the configuration $(x,y)$ and in the phase $(x,\dot{x})$ space in order to elucidate the escape process as well as the overall orbital properties of the galactic system. Our numerical analysis reveals the strong dependence of the properties of the considered escape basins with the total orbital energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. We hope our outcomes to be useful for a further understanding of the escape mechanism in active galaxy models., Comment: Published in Nonlinear Dynamics (NODY) journal. arXiv admin note: previous papers with related context: arxiv:1604.04622, arXiv:1505.03968, arXiv:1411.4864, arXiv:1508.05198, arXiv:1511.04908, arXiv:1511.04889, arXiv:1505.03847, arXiv:1511.04881, arXiv:1505.04185

The aim of this work is to compare the orbital dynamics in three different models describing the properties of a star cluster rotating around its parent galaxy in a circular orbit. In particular, we use the isochrone and the Hernquist potentials to model the spherically symmetric star cluster and we compare our results with the corresponding ones of a previous work in which the Plummer model was applied for the same purpose. Our analysis takes place both in the configuration $(x,y)$ and in the phase $(x,\dot{x})$ space in order to elucidate the escape process as well as the overall orbital properties of the tidally limited star cluster. We restrict our investigation into two dimensions and we conduct a thorough numerical analysis distinguishing between ordered and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels above the critical escape energy. It is of particular interest to determine the escape basins towards the two exit channels (near the Lagrangian points $L_1$ and $L_2$) and relate them with the corresponding escape times of the orbits., Comment: Published in MNRAS journal. arXiv admin note: previous papers with related context: arXiv:1411.4864, arXiv:1505.03968, arXiv:1505.03847

We explore the escape dynamics in open Hamiltonian systems with multiple channels of escape continuing the work initiated in Part I. A thorough numerical investigation is conducted distinguishing between trapped (ordered and chaotic) and escaping orbits. The determination of the location of the basins of escape towards the different escape channels and their correlations with the corresponding escape periods of the orbits is undoubtedly an issue of paramount importance. We consider four different cases depending on the perturbation function which controls the number of escape channels on the configuration space. In every case, we computed extensive samples of orbits in both the configuration and the phase space by numerically integrating the equations of motion as well as the variational equations. It was found that in all examined cases regions of non-escaping motion coexist with several basins of escape. The larger escape periods have been measured for orbits with initial conditions in the vicinity of the fractal structure, while the lowest escape rates belong to orbits with initial conditions inside the basins of escape. In addition, we related the model potential with applications in the field of reactive multichannel scattering. We hope that our numerical analysis will be useful for a further understanding of the escape mechanism of orbits in open Hamiltonian systems with two degrees of freedom., Published in Nonlinear Dynamics (NODY) journal. arXiv admin note: previous papers with related context: arXiv:1505.03968, arXiv:1411.4864, arXiv:1508.05198, arXiv:1404.4285

We numerically explore the Newton-Raphson basins of convergence, related to the libration points (which act as attractors), in the planar circular restricted five-body problem (CR5BP). The evolution of the position and the linear stability of the equilibrium points is determined, as a function of the value of the mass parameter. The attracting regions, on several types of two dimensional planes, are revealed by using the multivariate version of the classical Newton-Raphson iterative method. We perform a systematic investigation in an attempt to understand how the mass parameter affects the geometry as well as the degree of fractality of the basins of attraction. The regions of convergence are also related with the required number of iterations and also with the corresponding probability distributions., Published in Astrophysics and Space Science (A&SS) journal. Previous paper with related context: arXiv:1801.00710

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:1806.11409

The escape dynamics in a simple analytical gravitational model which describes the motion of stars in a Seyfert galaxy is investigated in detail. We conduct a thorough numerical analysis distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. In order to distinguish safely and with certainty between ordered and chaotic motion, we apply the Smaller ALingment Index (SALI) method. It is of particular interest to locate the escape basins through the openings around the collinear Lagrangian points $L_1$ and $L_2$ and relate them with the corresponding spatial distribution of the escape times of the orbits. Our exploration takes place both in the physical $(x,y)$ and in the phase $(x,\dot{x})$ space in order to elucidate the escape process as well as the overall orbital properties of the galactic system. Our numerical analysis reveals the strong dependence of the properties of the considered escape basins with the total orbital energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. It was also observed, that for energy levels close to the critical escape energy the escape rates of orbits are large, while for much higher values of energy most of the orbits have low escape periods or they escape immediately to infinity. We also present evidence obtained through numerical simulations that our model can describe the formation and the evolution of the observed spiral structure in Seyfert galaxies. We hope our outcomes to be useful for a further understanding of the escape mechanism in galaxies with active nuclei., Published in Nonlinear Dynamics (NODY) journal. arXiv admin note: previous papers with related context: arXiv:1411.4864, arXiv:1404.4285, arXiv:1505.03847

The H\'enon-Heiles potential was first proposed as a simplified version of the gravitational potential experimented by a star in the presence of a galactic center. Currently, this system is considered a paradigm in dynamical systems because despite its simplicity exhibits a very complex dynamical behavior. In the present paper, we perform a series expansion up to the fifth-order of a potential with axial and reflection symmetries, which after some transformations, leads to a generalized H\'enon-Heiles potential. Such new system is analyzed qualitatively in both regimes of bounded and unbounded motion via the Poincar\'e sections method and plotting the exit basins. On the other hand, the quantitative analysis is performed through the Lyapunov exponents and the basin entropy, respectively. We find that in both regimes the chaoticity of the system decreases as long as the test particle energy gets far from the critical energy. Additionally, we may conclude that despite the inclusion of higher order terms in the series expansion, the new system shows wider zones of regularity (islands) than the ones present in the H\'enon-Heiles system., Comment: 7 pages, 5 figures

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:1709.06631

The basins of convergence, associated with the roots (attractors) of a complex equation, are revealed in the Hill problem with oblateness and radiation, using a large variety of numerical methods. Three cases are investigated, regarding the values of the oblateness and radiation. In all cases, a systematic and thorough scan of the complex plane is performed in order to determine the basins of attraction of the several iterative schemes. The correlations between the attracting domains and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly suggests that the basins of convergence, with the highly fractal basin boundaries, produce extraordinary and beautiful formations on the complex plane., Published in Astrophysics and Space Science (A&SS) journal

The aim of this work is to revise but also explore even further the escape dynamics in the H\'{e}non-Heiles system. In particular, we conduct a thorough and systematic numerical investigation distinguishing between trapped (ordered and chaotic) and escaping orbits, considering only unbounded motion for several energy levels. It is of particular interest, to locate the basins of escape towards the different escape channels and relate them with the corresponding escape periods of the orbits. In order to elucidate the escape process we conduct a thorough investigation in several types of two-dimensional planes and also in a three-dimensional subspace of the entire four-dimensional phase space. We classify extensive samples of orbits by integrating numerically the equations of motion as well as the variational equations. In an attempt to determine the regular or chaotic nature of trapped motion, we apply the SALI method, as an accurate chaos detector. It was found, that in all studied cases regions of non-escaping orbits coexist with several basins of escape. Most of the current outcomes have been compared with previous related work., Comment: Published in Meccanica journal. arXiv admin note: Previous papers with related context: arXiv:1511.04908, arXiv:1511.04889, arXiv:1505.03847, arXiv:1609.00681, arXiv:1706.07298, arXiv:1604.04613, arXiv:1411.4864, arXiv:1508.05198, arXiv:1505.03968, arXiv:1604.04622

The case of the planar circular restricted three-body problem where one of the two primaries has a stronger gravitational field with respect to the classical Newtonian field is investigated. We consider the case where two primaries have the same mass, so as the the only difference between them to be the strength of the gravitational field which is controlled by the power $p$ of the potential. A thorough numerical analysis takes place in several types of two dimensional planes in which we classify initial conditions of orbits into three main categories: (i) bounded, (ii) escaping and (iii) collision. Our results reveal that the power of the gravitational potential has a huge impact on the nature of orbits. Interpreting the collision motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We successfully located the escape as well as the collision basins and we managed to correlate them with the corresponding escape and collision time of the orbits. We hope our contribution to be useful for a further understanding of the escape and collision properties of motion in this interesting version of the restricted three-body problem., Published in Meccanica journal. Previous papers with related context: arXiv:1512.08676, arXiv:1508.05209, arXiv:1608.08617, arXiv:1505.04185, arXiv:1511.04881, arXiv:1604.03403

The case of the classical Hill problem is numerically investigated by performing a thorough and systematic classification of the initial conditions of the orbits. More precisely, the initial conditions of the orbits are classified into four categories: (i) non-escaping regular orbits; (ii) trapped chaotic orbits; (iii) escaping orbits; and (iv) collision orbits. In order to obtain a more general and complete view of the orbital structure of the dynamical system our exploration takes place in both planar (2D) and the spatial (3D) version of the Hill problem. For the 2D system we numerically integrate large sets of initial conditions in several types of planes, while for the system with three degrees of freedom, three-dimensional distributions of initial conditions of orbits are examined. For distinguishing between ordered and chaotic bounded motion the Smaller ALingment Index (SALI) method is used. We managed to locate the several bounded basins, as well as the basins of escape and collision and also to relate them with the corresponding escape and collision time of the orbits. Our numerical calculations indicate that the overall orbital dynamics of the Hamiltonian system is a complicated but highly interested problem. We hope our contribution to be useful for a further understanding of the orbital properties of the classical Hill problem., Published in Nonlinear Dynamics (NODY) journal

The circular version of the restricted three-body problem, along with the method of grid classification are used to determine the character of the trajectories of a test particle, which move in a binary exoplanetary system. The binary system can be either a parent star-exoplanet or an exoplanet–exoplanet or exomoon, while the test particle is considered to be an asteroid or comet, a space probe, or even a small exomoon in the case where the primary body is a star. By using modern two-dimensional color maps, we succeeded in classifying the starting conditions and distinguishing between bounded, escaping, and collision type of motion for the test particle. Furthermore, in the case of bounded regular motion, we further classify the starting conditions by considering their geometry (revolving around one or both main bodies) and orientation (prograde or retrograde, with respect to a rotating coordinate system of the primaries). For the initial setup of the test particle we consider two starting conditions: the launch from pericenter or apocenter. The final states are qualitatively visualized through two-dimensional basin diagrams. This approach allowed us to systematically investigate and extract dynamical information on the dependency of the test particle final state as a function of the particle’s initial semi-major axis and eccentricity for a given primary and secondary mass ratio. Finally, we applied the restricted three-body model on several exoplanetary systems with observed mass-ratios and studied the dynamical behavior of a test-mass.

We examine the convergence properties of the generalized Hénon–Heiles system, by using the multivariate version of the Newton–Raphson iterative scheme. In particular, we numerically investigate how the perturbation parameter [Formula: see text] influences several aspects of the method, such as its speed and efficiency. Color-coded diagrams are used for revealing the basins of convergence on the configuration plane. Additionally, we compute the degree of fractality of the convergence basins on the configuration space, as a function of the perturbation parameter, by using different tools, such the uncertainty dimension and the (boundary) basin entropy. Our analysis suggests that the perturbation parameter strongly influences the number of the equilibrium points, as well as the geometry and the structure of the associated basins of convergence. Furthermore, the highest degree of fractality, along with the appearance of nonconverging points, occur near the critical values of the perturbation parameter.

We determine the character of orbits of stars moving in the meridional plane $(R,z)$ of an axially symmetric time-independent disk galaxy model with a spherical central nucleus. In particular, we try to reveal the influence of the value of the angular momentum on the different families of orbits of stars, by monitoring how the percentage of chaotic orbits, as well as the percentages of orbits of the main regular resonant families evolve when angular momentum varies. The smaller alignment index (SALI) was computed by numerically integrating the equations of motion as well as the variational equations to extensive samples of orbits in order to distinguish safely between ordered and chaotic motion. In addition, a method based on the concept of spectral dynamics that utilizes the Fourier transform of the time series of each coordinate is used to identify the various families of regular orbits and also to recognize the secondary resonances that bifurcate from them. Our investigation takes place both in the physical $(R,z)$ and the phase $(R,\dot{R})$ space for a better understanding of the orbital properties of the system. Our numerical computations reveal that low angular momentum stars are most likely to move in chaotic orbits, while on the other hand, the vast majority of high angular momentum stars perform regular orbits., Comment: Published in Mechanics Research Communications (MRC) journal. arXiv admin note: previous papers with related context: arXiv:1406.0446, arXiv:1404.4194

Understanding the regular or chaotic nature of orbits in galaxies is undoubtedly an issue of great importance. We determine the character of orbits of stars moving in the meridional plane (R, z) of an axially symmetric time-independent galactic model with a spherical central nucleus, and a flat biaxial oblate dark matter halo component. In particular, we try to reveal the influence of the fractional portion of dark matter on the structure and also on the different families of orbits of the galaxy, by monitoring how the percentage of chaotic orbits, as well as the percentages of orbits of the main regular resonant families evolve when the ratio of dark matter to luminous mass varies. The smaller alignment index (SALI) was computed by numerically integrating the equations of motion as well as the variational equations to extensive samples of orbits in order to distinguish between ordered and chaotic motion. In addition, a method based on the concept of spectral dynamics that utilizes the Fourier transform of the time series of each coordinate is used to identify the various families of regular orbits and also to recognize the secondary resonances that bifurcate from them. The investigation is carried out both in the physical (R, z) and the phase \((R,\dot {R})\) space for a better understanding of the orbital properties of the system. The numerical computations reveal that in both cases, the fractional portion of dark matter influences more or less, the overall orbital structure of the system. It was observed however, that the evolution of the percentages of all types of orbits as a function of the fractional portion of dark matter strongly depends on the particular type of space (physical or phase) in which the initial conditions of orbits are launched. The results are compared with the similar earlier work.