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• The photogravitational restricted four-body problem in two different configurations: (I) the equilateral four-body configuration (II) the collinear four-body problem are discussed. • Parametric evolution of the libration points are discussed. • We unveil the Newton-Raphson basins of convergence, linked with the libration points. • The basin entropy is illustrated to estimate the degree of fractality of the basins of convergence. The present problem deals with the effect of the radiation pressure on the topology of the basins of convergence in the restricted problem of four bodies in two different configurations: (I) the equilateral four-body configuration (II) the collinear four-body problem. We have illustrated the basins of convergence linked to the in-plane as well as out-of-plane libration points by applying the Newton-Raphson multivariate iterative scheme, in both the configurations. Additionally, the basin entropy is illustrated to estimate the degree of fractality of the basins of convergence. [ABSTRACT FROM AUTHOR]

• In this paper, a delayed HIV-TB co-infection model has been formulated. • The existence and stability of equilibrium points have been shown. • The condition under which Hopf-bifurcation takes place has been determined. • Ideal timing of the combination of treatment for both diseases has been computed. In this paper, a non-linear mathematical model for HIV-TB co-infection has been proposed. A constant time delay parameter is introduced into the model which accounts for the delay in detection and the commencement of appropriate treatment for co-infected individuals. The basic reproduction number along with the location and existence of equilibrium points is computed. Local stability analysis of all the equilibrium points in the presence of delay parameter is performed, which gives certain conditions, under which Hopf-bifurcation takes place. Numerical simulations are performed with different set of parameters for each equilibrium point to explore the effect of delay in detection and treatment of the infected individuals. The numerical results have also been used to describe the rich dynamics generated by the system in the presence of delay. [ABSTRACT FROM AUTHOR]

• The spatial collinear restricted four-body problem has been studied. • The peripheral primaries are oblate/prolate spheroid. • The existence and stability of libration points are discussed. • Parametric variation of the regions of possible motions are illustrated. • We unveil the Newton–Raphson basins of convergence, linked with the libration points. In the present work a systematic study has been presented in the context of the existence of libration points, their linear stability, the regions of motion where the test particle can orbit and the domain of basins of convergence linked to libration points in the spatial configuration of the collinear restricted four-body problem with non-spherical primaries (i.e., the primaries are oblate or prolate spheroid). The parametric evolution of the positions of the libration points as function of the oblateness or prolateness parameters of the primaries and the stability of these points in linear sense are illustrated numerically. Moreover, the numerical investigation shows that the only libration points which lie on either of the axes are linearly stable for several combinations of the oblateness parameter and mass parameter whereas the non-collinear libration points are found linearly unstable, consequently unstable in nonlinear sense also, for studied value of mass parameter and oblateness/prolateness parameter. Moreover, the regions of possible motion are also depicted, where the infinitesimal mass is free to orbit, as function of Jacobian constant. In addition, the basins of convergence (BoC) linked to the libration points are illustrated by using the multivariate version of the Newton–Raphson (NR) iterative scheme. [ABSTRACT FROM AUTHOR]

Highlights • The spatial collinear restricted four-body problem with variable mass is investigated. • The positions and stability of the libration points are discussed. • The evolution of the regions of possible motion are discussed. Abstract The present paper deals with the spatial collinear restricted four-body problem within the frame of repulsive Manev potential (− 1 r + e r 2) , e > 0 and variable mass. We have revealed that how the parameters involved due to variable mass of the infinitesimal body influence the locations as well as number of libration points. Moreover, it is observed that the libration points exist only on the coordinate axes. The regions of possible motion are highly influenced by the Manev parameter, parameter occur due to variable mass of the infinitesimal body as well as by the Jacobian constant. Further, it is unveiled that the libration points are unstable for all the involved parameters. [ABSTRACT FROM AUTHOR]