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The research presented in this work focuses on investigating the characteristics of equilibrium points within the context of the photogravitational restricted three-body problem. Specifically, the study considers the influences of albedo effect, a straight segment, radiation pressure, as well as small perturbations in the Coriolis and centrifugal forces. The dynamical model proposed in this study reveals the presence of five equilibrium points. Among these, three are collinear with the centers of the massive bodies, while the remaining two are non-collinear. By examining various parameters involved in the system, the effect on the position of the equilibrium points is explored through numerical and graphical analysis. These investigations provide valuable insights into how the positions of these points are affected by changes in the parameters. Another significant aspect addressed in this research is the stability of the equilibrium points. The study finds that the collinear equilibrium points are unstable, irrespective of the parameters involved. On the other hand, the non-collinear points exhibit stability within a specific range of the mass parameter. The precise range of stability depends on the values of the other parameters present in the system. Additionally, the research delves into the study of the permissible and forbidden regions of motion for an infinitesimal body within the system. This analysis is conducted by considering different values of the Jacobian constant and it suggests that the Jacobian constant significantly influences the dynamics of the system and has a direct impact on the permissible regions of motion of the infinitesimal body. [ABSTRACT FROM AUTHOR]

This study analyzes the motion of two infinitesimal bodies in the restricted problem of bodies, including an oblate spheroid and straight segment as primary bodies, along with the photogravitational effect of the more massive primary. The pertinent equations of motion of the infinitesimal bodies in the dimensionless variables are obtained by incorporating the effect caused by the oblateness, radiation pressure, and straight segment. On close examination of the parameters involved in the problem, a study of the equilibrium points of the dynamical system reveals that six of them are collinear with primary bodies' centers, while the remaining eight are not. An in-depth study is done on how the position of the equilibrium points changes with respect to the radiation parameter. The permissible and forbidden regions of motion are shown, and it is inferred that when the Jacobian constant is decreased, the permissible regions of motion expand. Furthermore, a linear stability analysis of the generated equilibrium points is done, and we observe that all the equilibrium points are unstable for a set of values of the model's parameters. [ABSTRACT FROM AUTHOR]

The present study investigates the combined effects of the oblateness and straight segment on the positions and linear stability of the equilibrium points in the restricted 2 + 2 body problem. The present model holds fourteen equilibrium points, out of which six are collinear with the centers of the primaries and the rest are non-collinear. It has been observed that the positions of all the equilibrium points are subsequently affected by the oblateness and length of the primary bodies. The linear stability of the equilibrium points is also presented by slightly perturbing the position of the equilibrium points. It is observed that for a considered set of parameters, all the fourteen equilibrium points are unstable. An application of the present model is also studied, for which the position and stability of the equilibrium points are investigated for the Earth-22 Kalliope-dual satellite system. It has been observed that for this system, all the equilibrium points are unstable except for two non-collinear equilibrium points that are found to be stable. [ABSTRACT FROM AUTHOR]

The present investigation deals with the outcomes of the small perturbations σ 1 and σ 2 in the Coriolis and centrifugal forces respectively, in the oblate straight segment setup of the restricted 2 + 2 body problem. We describe the dynamical system that consists of the primaries of masses m 1 and m 2 as oblate spheroid and straight segment, respectively. With this setup, we obtain the equations of motion of the infinitesimal bodies having masses m 3 and m 4 under the gravitational attraction of the primary bodies, their mutual attraction and the small perturbations in the Coriolis and centrifugal forces. We determine the equilibrium points by using the perturbation technique in the equilibrium points for the case m 4 = 0 . We get fourteen equilibrium points, six of which are on the line connecting the centers of the primaries and eight of which lie on the x y - plane but not on the x - axis. The stability of the equilibrium points for a particular set of the parameters is analyzed and it is concluded that the collinear equilibrium points are unstable. The four non-collinear equilibrium points are stable for some combinations of σ 1 and σ 2 and rest are unstable for all values of σ 1 and σ 2 . The permissible and forbidden regions of motion of the infinitesimal body are also studied for the different values of the Jacobian constant and it is perceived that the permissible regions of motion expands on decreasing the value of the Jacobian constant. These regions are subsequently affected by the involvement of the mass, oblateness, length and perturbations in the Coriolis and centrifugal forces parameters. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear population model for HIV-TB co-infection has been proposed. The model is incorporated with the effect of early and late initiation of HIV treatment in co-infectives already on TB treatment, on the occurrence of Immune Reconstitution Inflammatory syndrome (IRIS). A 15-dimensional (15D) mathematical model has been developed in this study. We begin with considering constant treatment rates and thereafter, proceed to time-dependent treatment rates for co-infectives as control parameters. The basic reproduction number, a threshold quantity, corresponding to each HIV and TB sub-model has been computed in case of constant controls. With constant values of control parameters, mathematical analysis shows the existence and local stability of the disease-free equilibrium point and the endemic equilibrium point for the model. Together with time-dependent parameters, an optimal control problem is introduced and solved using Pontryagin's maximum principle with an objective to minimize the number of infectives and disease induced deaths along with the cost of treatment. Numerical simulations are performed to examine the effect of reproduction numbers on control profiles and to identify, the ideal combination of treatment strategies which provides minimum burden on a society. Numerical results imply that if both HIV and TB are endemic in the population, then in order to bring in minimum burden from the co-infection, optimal control efforts must be enforced rather than constant treatment rate. [ABSTRACT FROM AUTHOR]

A non-linear SEIR mathematical model for coronavirus disease in India has been proposed, by incorporating the saturated incidence rate on the occurrence of new infections. In the model, the threshold quantity known as the reproduction number is evaluated which determines the stability of disease-free equilibrium and the endemic equilibrium points. The disease-free equilibrium point becomes globally asymptotically stable when the corresponding reproduction number is less than unity, whereas, if it is greater than unity then the endemic equilibrium point comes into existence, which is locally asymptotically stable under certain restrictions on the parameters value in the model. The impact of various parameters on the threshold quantity is signified by the sensitivity analysis. Numerical results imply that by implementing and strictly following the prevention measures a rapid reduction in the reproduction number for COVID-19 can be observed, through which the coronavirus disease can be controlled. [ABSTRACT FROM AUTHOR]

The fractal basins of convergence (BoC) linked with the equilibrium points are explored in the (N + 1)− body ring problem under the effect of small perturbations in the Coriolis and centrifugal forces when the bodies are sources of radiation. The evolution of the positions and the stability of the libration points and the possible regions of motion are determined as the function of mass parameter, the Coriolis, and centrifugal and radiation parameters. In addition, the parametric variation of the effect of the small perturbations in the Coriolis and centrifugal forces on the stability of the equilibrium points are also illustrated numerically. The multivariate version of the Newton–Raphson(NR) iterative method is used to analyze the effect of the radiation parameters and mass parameter on the topology of the BoC. [ABSTRACT FROM AUTHOR]

In the present paper, an effort to generalize the restricted problem of 2 + 2 bodies has been made by considering the primary bodies m1 and m2 to be a point mass and straight segment of length 2l, respectively. The motion of the two infinitesimal bodies mi, i = 3, 4 has been investigated, in which each mi, i = 3, 4 is moving in the gravitational field of mj, j = 1, 2, 3, 4 with i ≠ j. For the present problem, 14 equilibrium points are obtained, of which 6 are collinear and the rest are noncollinear. The length parameter l has a subsequent effect on the location of all the equilibrium points. Systematic stability analysis has been carried out by using the theory of perturbation. All the equilibrium points are found to be unstable. [ABSTRACT FROM AUTHOR]

The present paper deals with the regions of motion and the out-of-plane equilibrium points in the planar restricted three-body problem when both the primaries are heterogeneous oblate spheroids with three layers of different densities and sources of radiation. We have derived the equations of motion and examined the combine effects of radiation and the effect due to the different layers of primaries on the out-of-plane equilibrium points and regions of motion. It is further observed that the energy constant, the radiation parameter and the effect due to the different layers of primaries have substantial impact on the locations of the out-of-plane equilibrium points and on the regions of motion. Furthermore, we have observed that the out-of-plane equilibrium points are unstable for all mass ratios and other parameters, whereas the infinitesimal mass can move freely around the primaries for smaller values of Jacobi constant. But, when we increase the radiation parameter due to either of the primaries, the forbidden regions increase, where the motion is not possible. Moreover, we have discussed the Newton-Raphson basins of attraction to observe the effect of various perturbation on the topology of basins of convergency on configuration plane. [ABSTRACT FROM AUTHOR]

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

In this paper, we have studied the effect of oblateness of the Earth on the resonant curve in synchronous satellite under the gravitational effect of the Sun, the Moon and the Earth. By following the procedure of Frick and Garber (1962), we obtained the equations of motion of the synchronous satellite in the form of a second order linear differential equation by using the perturbation technique. From the solution, we observed that oscillatory amplitude becomes infinitely large when the ratio of two frequencies (angular velocity of the bary-center system around the Sun) and (spin rate of Earth) are commensurable. With the help of different graphs, we have shown the effect of and (coefficient of Earth's oblateness) on the resonant curves. It is observed that oscillatory amplitude increases when (orbital angle of the bary-center system around the Sun) increases and the value of is very close to zero. We also analyzed the phase portrait and phase space by using the method of Poincare section when the system is free from all the forces. Finally, we have obtained the expression for the energy integral, motion of mean longitude near the synchronous satellite. [ABSTRACT FROM AUTHOR]

The second wave of COVID-19 is an unprecedented condition in India and began in mid February 2021. Individuals who were already suffering from other comorbidities were found with lung infection, and hence, the number of disease induced deaths were rising faster during the second wave in relation to the first wave. This paper has proposed a mathematical model with fractional order derivatives by correlating the model based number of infectives with the real number of infectives in India. For the system of fractional differential equations, a disease-free state has been computed and proved to be locally asymptotically stable with certain restrictions. The mathematical model has been numerically simulated using the predictor-corrector method to highlight the role played by fractional order in controlling the disease spread. Numerical simulations signify the fact that a vital role has been played by fractional order model over integer order model in determining the transmission of COVID-19. It can be visualized that the increment rate in the infectives is lower by taking into consideration the memory effect due to a previous exposure to COVID-19. [ABSTRACT FROM AUTHOR]

The effects of the third-order terms ɛ and η in the effective potential V = 1 2 (A x 2 + B y 2) − ɛ x 2 y − η y 3 , on the topology of Newton–Raphson basins of convergence (NRBoC), linked to the equilibrium points are presented. The evolution of the positions of the libration points as a function of the parameters ɛ and η is determined. The basins of convergence (BoC) in (x , y) plane, are unveiled by using the multivariate version of the Newton–Raphson (N-R) iterative scheme. In an attempt to analyse the effect of third order terms on the topology of the BoC as well as the basin entropy, we have performed a systematic numerical investigation. It is revealed that the third-order terms ɛ and η have significant effects on the topology of the BoC linked to the libration points. • The Hénon–Heiles type system has been studied. • The positions of libration points are illustrated. • The basins of convergence are unveiled. • The basins entropy are presented. [ABSTRACT FROM AUTHOR]

We have investigated the resonances due to the perturbations of a geo-centric synchronous satellite under the gravitational forces of the Sun, the Moon and the Earth including it's equatorial ellipticity. The resonances at the points resulting from (i) the commensurability between $\dot{\theta}_{0}$ (steady-state orbital angular rate of the satellite) and $\dot{\theta}_{m}$ (angular velocity of the moon around the earth) and (ii) the commensurability between $\dot{\theta}_{0}$ and $\dot{\psi}_{0}$ (steady-state regression rate of the synchronous satellite) are analyzed. The amplitude and the time period of the oscillation have been determined by using the procedure as given in Brown and Shook (Planetary Theory, Cambridge University Press, Cambridge, ). We have observed that as θ (0≤ θ≤45) and ψ (0≤ ψ≤135) increase, the amplitude decreases and the time period also decreases. We have also shown the effect of ψ on amplitude and time period for 0≤ Γ≤45, where Γ is the angle measured from the minor axis of the earth's equatorial ellipse to the projection of the satellite on the plane of the equator. [ABSTRACT FROM AUTHOR]

In this problem, one of the primaries of mass $m^{*}_{1}$ is a rigid spherical shell filled with a homogeneous incompressible fluid of density ρ. The smaller primary of mass m is an oblate body outside the shell. The third and the fourth bodies (of mass m and m respectively) are small solid spheres of density ρ and ρ respectively inside the shell, with the assumption that the mass and the radius of the third and the fourth body are infinitesimal. We assume that m is describing a circle around $m^{*}_{1}$. The masses m and m mutually attract each other, do not influence the motions of $m^{*}_{1}$ and m but are influenced by them. We also assume that masses m and m are moving in the plane of motion of mass m. In the paper, equilibrium solutions of m and m and their linear stability are analyzed. There are two collinear equilibrium solutions for the system. The non collinear equilibrium solutions exist only when ρ= ρ. There exist an infinite number of non collinear equilibrium solutions of the system, provided they lie inside the spherical shell. In a system where the primaries are considered as earth-moon and m, m as submarines, the collinear equilibrium solutions thus obtained are unstable for the mass parameters μ, μ, μ and oblateness factor A. In this particular case there are no non-collinear equilibrium solutions of the system. [ABSTRACT FROM AUTHOR]

In non-linear stability analysis, the effects of non-linear terms in the equations of motion are considered. The presence of these non-linear terms can cause the results of linear stability to change dramatically. Therefore, the Arnold–Moser theorem (Kolmogorov–Arnold–Moser theory) has been used to study the non-linear stability of the equilibrium point in the planar Robe's restricted three-body problem when the second primary is a finite straight segment of length . The density parameter is considered zero. The equilibrium point has been found to be stable in non-linear sense for all mass ratios in the range of linear stability except possibly for six critical values of the mass ratio . The critical values depend on the length parameter , which shows that the length parameter has significant effect on the non-linear stability of the equilibrium point. The obtained results are applied to predict the stability of equilibrium point for Jupiter–Amalthea system. It is observed that the equilibrium point is unstable for Jupiter‒Amalthea system. [ABSTRACT FROM AUTHOR]

In the framework of axisymmetric problem of restricted five bodies with variable mass, our goal is to study the existence and stability of libration points. In the axisymmetric restricted five-body problem, we have assumed that the mass of the fifth particle varies according to Jeans' law. We have further supposed that the four bodies having masses and (with ) form an axisymmetric concave configuration. The equations of motion of a test particle of infinitesimal mass have been illustrated, which is similar to the axisymmetric restricted five-body problem when the problem of variable mass evolves with the problem of constant mass. In this paper, we have determined in-plane as well as out-of-plane libration points along with their stability. Further, we have shown the regions of motion where the fifth body can move freely. We have also shown that the angle parameters and and the parameters, arising due to variation of mass and , affect the existence and number of these libration points. Moreover, the bivariate version of the Newton-Raphson iterative scheme is applied in an attempt to unveil the analysis of the basins of convergence linked with the libration points as a function of the mass parameter. [ABSTRACT FROM AUTHOR]

Resonances in a geo-centric synchronous satellite under the gravitational forces of the Sun, the Moon and the Earth including it's equatorial ellipticity have been investigated. The resonance at two points resulting from the commensurability between the mean motion of the satellite and Γ (angle measured from the minor axis of the Earth's equatorial ellipse to the projection of the satellite on the plane of the equator) is analyzed. The amplitude and the time period of the oscillation have been determined by using the procedure of Brown and Shook. We have observed that the amplitude and the time period of the oscillation decrease as Γ increases in the first quadrant. The radial deviation (Δ r) and the tangential deviation ( rΔ θ) have been determined. Here r represents the synchronous altitude. The effects of the arithmetic sum of amplitudes λ involved in the perturbation equations on orbital inclination 0≤ α≤90 are shown. It is observed that $\sum_{i = 1}^{46} \lambda_{i}$ increases as α increases. We have also determined the displacement Δ D (called drift) due to the oscillatory terms under the summation sign involved in the equations of motion of the satellite. We have observed that the value of Δ D is less than 0.5. [ABSTRACT FROM AUTHOR]

In this problem, one of the primaries of mass m1 is a Roche ellipsoid filled with a homogeneous incompressible fluid of density ρ1. The smaller primary of mass m2 is an oblate body outside the Ellipsoid. The third and the fourth bodies (of mass m3 and m4 respectively) are small solid spheres of density ρ3 and ρ4 respectively inside the Ellipsoid, with the assumption that the mass and the radius of the third and the fourth body are infinitesimal. We assume that m2 is describing a circle around m1. The masses m3 and m4 mutually attract each other, do not influence the motions of m1 and m2 but are influenced by them. We have extended the Robe’s restricted three-body problem to 2+2 body problem under the assumption that the fluid body assumes the shape of the Roche ellipsoid (Chandrashekhar in Ellipsoidal figures of equilibrium, Chap. 8, Dover, New York, 1987 ). We have taken into consideration all the three components of the pressure field in deriving the expression for the buoyancy force viz (i) due to the own gravitational field of the fluid (ii) that originating in the attraction of m2 (iii) that arising from the centrifugal force. In this paper, equilibrium solutions of m3 and m4 and their linear stability are analyzed. We have proved that there exist only six equilibrium solutions of the system, provided they lie within the Roche ellipsoid. In a system where the primaries are considered as Earth-Moon and m3, m4 as submarines, the equilibrium solutions of m3 and m4 respectively when the displacement is given in the direction of x1-axis or x2-axis are unstable. [ABSTRACT FROM AUTHOR]

We have investigated the resonances in the earth-moon system around the sun including earth's equatorial ellipticity. The resonance resulting from the commensurability between the mean motion of the moon and Γ (angle measured from the minor axis of the earth's equatorial ellipse to the projection of the moon on the plane of the equator) is analyzed. The amplitude and the time period of the oscillation have been determined by using the procedure of Brown and Shook. We have shown the effects of Γ on the amplitude and the time period of the resonance oscillation using the data of the moon. It is observed that the amplitude decreases and the time period also decreases as Γ increases from 0 to 45. [ABSTRACT FROM AUTHOR]

The resonances in a geocentric satellite due to earth's equatorial ellipticity have been investigated. The resonance at five points resulting from the commensurability between the mean motion of the satellite and the earth's equatorial ellipticity is analyzed. The amplitude and the time period of the oscillation have been determined by using the procedure of Brown and Shook. A comparison of their effects on the orbital elements has also been studied. It is observed that the amplitude and the time period of the oscillation decrease as Γ (angle measured from the minor axis of the earth's equatorial ellipse to the projection of the satellite on the plane of the equator) increases in the first quadrant for all the resonance cases. [ABSTRACT FROM AUTHOR]

• The modified circular restricted three-body problem has been studied. • The parametric evolution of the stationary points are illustrated. • The effect of the Coriolis and centrifugal force on the stationary points and their stability are unveiled. • The multivariate version of the NR-iterative scheme is used to discuss the basins of convergence. In the present manuscript, the effects of small perturbations in the centrifugal and Coriolis forces with additional effect of three-body interaction on the locations as well as stability of the stationary points (SPs), and on the topology of the basins of convergence (BoCs) linked with these SPs in the restricted three-body problem (R3BP) are discussed by using the multivariate version of the Newton-Raphson iterative scheme. It is found that the positions of the SPs are highly influenced by the centrifugal force but their stability is highly influenced by combinations of the centrifugal and Coriolis forces. In addition, the correlations betwixt the basin of convergence (BoC) connected to the SPs and the associated number of iterations needed to assure pre-assumed precision are analyzed. Our analysis suggests that the mass ratio μ , three-body interaction parameter k and the effect in centrifugal and the Coriolis forces controlled by parameters ϕ and φ are very influential parameters. [ABSTRACT FROM AUTHOR]

The first Tension Leg Platform (TLP) was installed in 1984 to develop the Hutton field in the Central North Sea in about 500 ft of water. It successfully demonstrated the ability of a floating platform tethered to the seabed to drill and produce with surface trees. In the ensuing years, twenty three additional TLPs have been installed and five more sanctioned, in most major deepwater producing regions around the world, in water depths down to 5,200 ft. The TLP today is a mature and proven deepwater production platform and is routinely included as a platform concept building block for many deepwater prospects during field development planning for dry or wet tree scenarios. This paper presents a retrospective of TLP development that includes: An overview and description of TLP hull configurations Basic hull and tendon design considerations A chronology of TLP designs, applications and contracting strategies The evolution of TLP tendon and foundation systems [ABSTRACT FROM AUTHOR]

This research article presents the effect of the Earth's equatorial ellipticity and the resistive force on a geocentric satellite's resonant curves. The resonance occurs in the motion of the satellite due to the frequencies, the angular velocity of the satellite, the rate of change of Earth's equatorial ellipticity parameter (γ ̇) , the steady-state angular velocity of the Moon around the Earth ( ϑ 0 ̇) , and the steady-state angular velocity of barycenter (Θ ̇) around the Sun. We express the equations of motion of the satellite in a spherical coordinate system by using the Earth's potential. By applying the unperturbed solution, we reduced the system of equations into a second-order differential equation and obtained its solution. From the solution it is observed that the resonance appears between the frequencies δ s 0 ̇ , γ ̇ , Θ 0 ̇ and ϑ 0 ̇ at ten resonant points. We analyze the resonant curves corresponding to the resonant points and show the effect of resistive force and latitude of a satellite. Also, We show the effect of eccentricity (e) and semi-major axis (a) on the oscillatory amplitudes (A i) , for i = 1 , 2 , 3 ... 21. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new dynamical model addressing the variation in climate condition due the presence of microorganisms. We also introduce a new dynamical model of cancer growth which includes three interactive cell populations with drug free environment, namely tumor cells, healthy host cells, and immune effector cells. In this, we considered the super growth of tumor cells. For the choice of certain parameters, both of the systems exhibit chaotic behavior. The aim of this work is to design the controller to control the chaos and to provide sufficient conditions which achieve synchronization of two non-identical systems, which is based on Lyapunov stability theory. To verify synchronization is achieved between the systems, we performed the numerical simulation. [ABSTRACT FROM AUTHOR]

In the problem of 2+2 bodies in the Robe's setup, one of the primaries of mass $m^{*}_{1}$ is a rigid spherical shell filled with a homogeneous incompressible fluid of density ρ. The second primary is a mass point m outside the shell. The third and the fourth bodies (of mass m and m respectively) are small solid spheres of density ρ and ρ respectively inside the shell, with the assumption that the mass and the radius of third and fourth body are infinitesimal. We assume m is describing a circle around $m^{*}_{1}$. The masses m and m mutually attract each other, do not influence the motion of $m^{*}_{1}$ and m but are influenced by them. We also assume masses m and m are moving in the plane of motion of mass m. In the paper, the equations of motion, equilibrium solutions, linear stability of m and m are analyzed. There are four collinear equilibrium solutions for the given system. The collinear equilibrium solutions are unstable for all values of the mass parameters μ, μ, μ. There exist an infinite number of non collinear equilibrium solutions each for m and m, lying on circles of radii λ, λ′ respectively (if the densities of m and m are different) and the centre at the second primary. These solutions are also unstable for all values of the parameters μ, μ, μ, φ, φ′. Such a model may be useful to study the motion of submarines due to the attraction of earth and moon. [ABSTRACT FROM AUTHOR]

The modified circular restricted three-body problem is numerically investigated by exploring the effect of three-body interaction on the basins of convergence connected to the in-plane as well as out-of-plane equilibrium points. The evolution of the positions of the libration points and their stability are illustrated as a function of parameter k due to three-body interaction. It is observed that the number of equilibrium points strongly depends on the sign and the magnitude of the three-body interaction parameter and there exist at most seven libration points where four are non-collinear and three are collinear with the primaries. Moreover, in the Copenhagen case we have found seven collinear libration points. Moreover, the non-collinear as well as collinear libration points are stable for various combinations of mass parameter μ and k. The collinear as well as non-collinear libration points are stable for even those values of μ which are higher than the μ crit of the classical restricted three-body problem. The study of the regions of possible motion shows that as the value of the Jacobian constant decreases, the forbidden region decreases significantly. The attracting domains of the basins of convergence, on several types of two-dimensional planes, are unveiled by applying the multivariate Newton-Raphson iterative scheme. In an attempt to analyse the effect of parameter k on the topology of basins of convergence, a systematic and thorough investigations are presented. The degree of fractality is also unveiled by determining the basin entropy of the convergence plane. • The restricted three-body problem with additional effect of three-body interaction is discussed. • The existence and stability of libration points are discussed. • We numerically investigated the topology of BoCs. • The basin entropy S_b as function of three-body interaction parameter are unveiled. [ABSTRACT FROM AUTHOR]

We have investigated the non-linear stability of the triangular libration point L4 of the Restricted three body problem under the presence of the third and fourth order resonances, when the bigger primary is an oblate body and the smaller a triaxial body and both are source of radiation. It is found through Markeev's theorem that L4 is always unstable in the third order resonance case and stable or unstable in the fourth order resonance case depending upon the values of the parameters A1, A′1, A′2, P and P′, where A1, A′1 and A′2, depends upon the lengths of the semi axes of the primaries and P and P′ are the radiation parameters. [ABSTRACT FROM AUTHOR]

• The regular polygon problem of (N + 1) -bodies with spheroidal primaries are discussed. • Existence and stability of the libration points are discussed. • We unveil the Newton-Raphson basins of convergence, linked to the libration points. • The parametric evolution of the basin entropy is presented. In the present manuscript, we unveil the topology of the basins of convergence in the regular polygon problem of (N + 1) -bodies in two different cases, i.e., in case-I, only the central primary creates the Manev-type quasi-homogeneous potential, and in case-II, the peripheral primaries create the Manev-type quasi-homogeneous potential. The regular polygon problem of (N + 1) -bodies describes the motion of the test particle moving in the force field of N primaries, the ν = N − 1 peripheral primaries of equal masses situated at the vertices of the imaginary regular ν -gon and the Nth primary with different mass i.e., the central primary situated at the centre of mass of the system. In this model, we assume that the primaries create quasi-homogeneous potentials instead of Newtonian potentials and forces. In order to approximate various phenomena due to the irregular shape of the primaries or due to emitting radiation, an inverse cube corrective term is inserted to the inverse square Newtonian law of gravitation. We, numerically, investigated the evolution of the positions of the libration points and their linear stability for different values of ν in both the cases. Further, the multivariate version of Newton-Raphson iterative scheme is applied to unveil the topology of the basins of convergence. Moreover, the "basin entropy" is also computed to analyse the basins of convergence quantitatively. [ABSTRACT FROM AUTHOR]

The belief that celestial bodies as perfect spheres results in certain idealistic conditions because most often they are in irregular shapes. With this in mind, we conduct an analysis to study the restricted three-body problem by taking the primaries as oblate spheroids. The modified mean motion expression [1] is used by incorporating the secular perturbation effect due to oblateness of the primaries on mean anomaly, argument of perigee and right ascension of ascending node. The model possesses five equilibrium points, which are subsequently affected by the oblateness parameters. Furthermore, we have studied the stability of the equilibrium points and it is observed that the collinear equilibrium points are always unstable. However, the non-collinear equilibrium points are stable for some combinations of the involved parameters. We have also plotted the zero velocity curves of the infinitesimal body for different values of the Jacobian constant and oblateness parameter. It has been observed that the value of the Jacobian constant has been playing a vital role in obtaining the permissible regions of motion of the infinitesimal body. Further, the results obtained of the study are applied to study the motion of a satellite in the Saturn–Titan system. [ABSTRACT FROM AUTHOR]

• The modified R3BP with variable mass are discussed. • Existence and stability of the libration points are discussed. • The parametric evolution of the ZVCs as the function of parameter used is presented. • We unveil the Newton-Raphson basins of convergence, linked to the libration points. In the present work, we shall show an exhaustive numerical analysis of the dynamical behavior of the test particle in the modified restricted three-body problem with variable mass in which the extra effect of the three-body interaction is also taken into account. This additional force ingredient appears in the potential of the classical problem as a new extra term. As the main results, we determine the motion for test particle, under the effect of three-body interaction, which varies its mass according to Jeans' law (Jeans (1928)). The number and existence of the libration points along with their stability have been investigated as the function of value of the parameters which occur due to variable mass of the test particle. Moreover, the regions of the possible motion are also unveiled where the test particle is free to move. Furthermore, the multivariate version of the Newton-Raphson (NR) iterative scheme is used to determine the outcomes of the used parameters on the topology of the basins of convergence (BoC) linked to the libration points. The numerical analysis shows that the topology of the basins of convergence linked with the libration points is highly influenced by the used parameters. Moreover, we perform a systematic analysis to unveil how the regions of convergence are related with the number of required iterations and also with the corresponding probability distributions. [ABSTRACT FROM AUTHOR]

• The position of equilibrium points are monitored with respect to the parameters the distances between galaxies (R), the masses of disk and nucleus of galaxies (M d 1 , M n 1 , M d 2 , M n 2 ). • The zero velocity curves are investigated under the effect of above parameters. • We determine the Newton-Raphson's basins of attraction associated with the libration points. • The existence of unpredictable (fractal) regions are explored using the method of basin entropy. • Due to variation of the parameter R, the change in basin entropy (S b) and boundary basin entropy (S bb) is graphically shown. The present paper investigates the binary system of quasars in the framework of the Circular Restricted Three-Body Problem. The parametric evolution of libration points and the geometry of zero-velocity curves are one of the crucial aspects of our study. The multivariate form of NR method is applied to study the basin of attraction connected with libration points. The algorithm for using the Newton-Raphson method is slightly modified in order to avoid the unnecessary delay in the convergence of initial conditions. The impact of parameters on the pattern of the basin of attraction and the number of iterations needed for the convergence of initial conditions are explored. We carry out an exhaustive (numerical) study to show the influence of these parameters on converging regions in basins of attraction. We unveil the existence of fractal structure in the basin of attraction using the method of basin entropy. In almost all cases, the existence of fractal structure is found throughout the basins of attraction. [ABSTRACT FROM AUTHOR]

In this article, we investigated the resonant curves of geo-centric satellite due to the rate of change in earth's equatorial ellipticity parameter (γ ˙) , moon's angular velocity around the earth (ϑ ˙) , and angular velocity of bary-center (Θ ˙) around the sun. We expressed the equations of the motion of the satellite in a spherical coordinate system using the earth's potential. The system of equations are reduced into a second-order ordinary differential equation and determined its solution. From the solution, it is noted that resonance appears due to the frequencies δ s 0 ˙ , γ ˙ , Θ 0 ˙ and ϑ 0 ˙ at ten resonant points δ ˙ s 0 = 2 γ ˙ , δ ˙ s 0 = γ ˙ , δ ˙ s 0 = 2 Θ ˙ 0 , δ ˙ s 0 = Θ ˙ 0 , δ ˙ s 0 = 2 Θ ˙ 0 3 , δ ˙ s 0 = Θ ˙ 0 2 , δ ˙ s 0 = 2 ϑ 0 ˙ , δ ˙ s 0 = ϑ 0 ˙ , δ ˙ s 0 = 2 ϑ 0 ˙ 3 , and δ ˙ s 0 = d o t ϑ 0 2 , where δ ˙ s 0 , Θ ˙ 0 and ϑ 0 ˙ are the steady-state value of δ ˙ s , Θ ˙ and ϑ ˙ respectively. We have drawn the resonant curves corresponding to the resonant points. From the graph, it is observed that the effect of earth's equatorial ellipticity parameter on the resonant curves are very small. Also, we have shown the effect of eccentricity (e) and semi-major axes (a) on the oscillatory amplitudes (A i) , where i = 1 , 2 , 3 ... 21 . Finally, we have analyzed the phase portrait by using Poincaré section method when the system is free from forces. [ABSTRACT FROM AUTHOR]

In the present paper, a study on the beyond-Newtonian collinear circular restricted (3 + 1) -body problem with spinning primaries is presented. In the setup of the collinear circular restricted four-body problem (CCR4BP), the three primaries are situated in the collinear central configuration where the masses of two peripheral primaries are equal. We have first derived the equations of motion for a system composed of three Kerr-like primaries. Further, we have unveiled how the existence, evolution, and stability of the libration points in the CCR4BP with beyond-Newtonian first-order correction terms and spin of the primaries depend on the mass parameter β and control parameter ε intending to manage the contribution of the beyond-Newtonian terms. It is observed that equilibria of the system are always even and varying from six to eighteen. [ABSTRACT FROM AUTHOR]

In the present paper, we have determined the equations of motion of the Moon in spherical coordinate system using the gravitational potential of Earth. Using perturbation, equations of motion are reduced to a second order differential equation. From the solution, two types of resonance are observed: (i) due to the frequencies-rate of change of Earth's equatorial ellipticity parameter and Earth's rotation rate, and (ii) due to the frequencies-angular velocity of the bary-center around the sun and Earth's rotation rate. Resonant curves are drawn where oscillatory amplitude becomes infinitely large at the resonant points. The effect of Earth's equatorial ellipticity parameter and resistive force on the resonant curve is analyzed. From the graphs it is observed that the effect of Earth's equatorial ellipticity on the resonant curve is very small while the effect of resistive force is significant. It is also observed that oscillatory amplitude decreases when the magnitude of resistive force increases. Finally, the phase portrait is analyzed when the system is free from forces. Orbits in the phase space are also studied by applying the method of Poincare section. A necessary condition for the bifurcation is derived at the end. [ABSTRACT FROM AUTHOR]

Associating the influences of viscosity and oblateness in the finite straight segment model of the Robe's problem, the linear stability of the collinear and non-collinear equilibrium points for a small solid sphere are analyzed. This small solid sphere is moving inside the first primary which is a homogeneous incompressible viscous fluid whose hydrostatic equilibrium figure is an oblate spheroid. The second primary is a finite straight segment. The existence of the equilibrium points is discussed after deriving the pertinent equations of motion of the small solid sphere. It is found that viscosity does not affect the location and number of equilibrium points but affects the stability as it converts the marginal stability to asymptotic stability. However, oblateness affects the locations of the equilibrium points. Applicability of the results of this study to an astrophysical problem is discussed, and we have calculated a lower bound on the ratio of the orbital radius and the total mass of the primaries of an astrophysical problem to which the results obtained may be applied. This ratio is called the scaling factor. [ABSTRACT FROM AUTHOR]

The present study is motivated by the work of Abouelmagd et al. (Int J Geom Methods Mod Phys 18(4):21500055, 2020), that it is generalized by considering the effect of small perturbations in the Coriolis and centrifugal forces on the location and stability of the equilibrium points in the Robe's restricted three-body problem. The problem possesses a pair of collinear equilibrium points and an infinite number of non-collinear equilibrium points. All of the resulting equilibrium points are subjected to a linear stability study. It has been observed that the locations and stability are guaranteed to be influenced by the perturbations in the Coriolis and centrifugal forces. Furthermore, a numerical study has been carried out to observe how the perturbing parameters affect the position and stability of the equilibrium points. [ABSTRACT FROM AUTHOR]

The analysis of the affect of angular velocity on the geometry of the basins of convergence (BoC) linked to the equilibrium points in the restricted three-body problem is illustrated when the primaries are source of radiation. The bivariate scheme of the Newton–Raphson (N–R) iterative method has been used to discuss the topology of the basins of convergence. The parametric evolution of the fractality of the convergence plane is also presented where the degree of fractality is illustrated by evaluating the basin entropy of the convergence plane. • The perturbed restricted three-body problem has been studied. • Existence of the libration points are discussed, as a function of the angular velocity. • We unveil the Newton–Raphson basins of convergence, linked with the libration points. • Basin entropy as function of angular velocity and radiation parameter are illustrated. [ABSTRACT FROM AUTHOR]

In this paper, a HIV/AIDS and TB co-infection model has been explored, which incorporates detection and treatment for both diseases. We begin with presenting a co-infection model and then start analyzing both the HIV and the TB sub-models separately. The basic reproduction numbers corresponding to both HIV and TB are computed. Both the HIV/AIDS and the TB sub-models have been shown to exhibit two equilibrium points, namely, the disease-free equilibrium point and the unique endemic equilibrium point. For both sub-models, the disease-free equilibrium point is locally as well as globally asymptotically stable when their corresponding reproduction number is less than unity. The endemic equilibrium point for both sub-models exists, when their corresponding reproduction numbers is greater than one. We also analyze the full HIV-TB co-infection model. With the aim of minimizing infectives and the cost of applying effort towards the detection and the treatment, optimal control analysis is performed for the full model using the Pontryagin's maximum principle. Numerical simulations with different combinations of efforts are then performed to explore the effect of detection in the presence of treatment for both diseases. Numerical simulations emphasize the fact that to reduce co-infection from the population, programs to accelerate the detection of both diseases are also required along with the treatment. • This paper presents a non-linear HIV-TB co-infection model. • Local stability analysis for the model has been done. • An optimal control problem to minimize the total cost is presented. • The existence and characterization of the optimal controls has been established. [ABSTRACT FROM AUTHOR]

In the framework of photogravitational version of the restricted five-body problem, the existence and stability of the in-plane equilibrium points, the possible regions for motion are explored and analysed numerically, under the combined effect of small perturbations in the Coriolis and centrifugal forces. Moreover, the multivariate version of the Newton-Raphson iterative scheme is applied in an attempt to unveil the topology of the basins of convergence linked with the libration points as function of radiation parameters, and the parameters corresponding to Coriolis and centrifugal forces. [ABSTRACT FROM AUTHOR]

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

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

This article aims to numerically investigate the orbital dynamics of the planar circular‐restricted three‐body problem, where the two primary bodies have equal masses and an oblate spheroid shape. By numerically integrating several large sets of initial conditions of both retrograde and prograde orbits, we classify them into three main categories: (a) bounded (regular or chaotic), (b) escaping, and (c) collision orbits. We reveal 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 (or equivalently of the Jacobi constant) as well as of the oblateness coefficient. Our numerical results are compared with related previous ones, corresponding to the classical version of the planar‐restricted three‐body problem. The presented outcomes have practical physical applications as they can be used for modeling the motion of a massless test particle in the gravitational field of a binary system of two oblate exoplanets. [ABSTRACT FROM AUTHOR]

In this paper, we present the analysis of the Newton–Raphson basins of convergence (NR-BoCs) linked with the relative equilibria (LPs) in the Eulerian configuration of a pseudo-Newtonian planar circular restricted problem of four bodies (SCCR4BP) with spinning primaries. In this configuration of the SCCR4BP, the primaries are laid out in a symmetrical collinear shape in which the two peripheral primaries are of equal mass. We derive the equations of motion by assuming that the system consists of three Kerr-like primaries to examine how the geometry of the basins of convergence (BoCs) associated with the LPs of SCCR4BP are affected by the first-order non-Newtonian contribution to the gravitational field. Additionally, we analyze the correlation among the regions of convergence, the required number of iterations, and the associated probability distributions. Finally, for the quantitative study of the basins of convergence, we measure the uncertainty of the respective basins via the basin entropy and the basin boundary entropy. • On the beyond-Newtonian collinear circular restricted (3 + 1)-body problem with spinning primaries is discussed. • The Newton–Raphson basins of convergence are discussed. • The basin entropy and basin boundary entropy are calculated to measure the uncertainty of the basins. [ABSTRACT FROM AUTHOR]

Highlights • The restricted five-body problem with variable mass has been studied. • The locations of the libration points and their stability have been discussed. • The basins of convergence linked with the libration points are discussed. • The Newton–Raphson iterative scheme is used. Abstract In the framework of restricted five-body 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 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 basins domain of convergence associated with the libration points are substantially influenced by the perturbation parameter. [ABSTRACT FROM AUTHOR]

Highlights • The autonomous restricted four-body problem with variable mass has been studied. • The locations of libration points and their stability have been discussed. • The effect of parameter as well as Jacobian constant on the regions of possible motion are unveiled. • The basins of convergence associated with the libration points using Newton–Raphson iterative scheme are discussed. Abstract We have numerically investigated the circular autonomous restricted four-body problem where the fourth particle of variable mass is moving under the gravitational influence of three bodies known as primaries. Moreover, these primaries move in circular orbit around their common center of mass in such a way that their configuration remains an equilateral triangle configuration. The effect of the parameter α on the existence as well as on the locations of the libration points are investigated. The parametric variation of the positions of the libration points and zero velocity curves are also revealed when the parameter α (which occurs in Jeans' law) increases. Moreover, the Newton–Raphson basins of convergence corresponding to the libration points are unveiled numerically when the parameter α increases. The obtained results strongly suggest that the study of the evolution of the attracting domains of the proposed dynamical system is worth studying in spite of their complexity. [ABSTRACT FROM AUTHOR]

Highlights • The restricted four-body problem with pseudo-Newtonian potential is investigated. • The position and stability of the libration points are discussed. • The number of libration points depend on transition parameter. • The evolution of the region of possible motion are discussed. • The Newton-Raphson iterative scheme is used to unveil the basins of convergence. Abstract 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 ϵ ∈(0, 1]. The stability of these libration points has also been discussed. Our study reveals that the Jacobi constant as well as transition parameter ϵ 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. [ABSTRACT FROM AUTHOR]