Effect of Variable Mass on N–R Basins of Convergence in Photogravitational Magnetic Binary Problem

Abstract

In this paper, we consider the photogravitational magnetic binary problem (PMBP) in the framework of a restricted three-body problem. In this model, the mass of a third body is variable and both primaries are considered as magnetic dipoles. Additionally, the effect of gravitational forces of both primaries is also considered. The effect of variable mass parameters $\alpha$ and $\beta$ on the existence and evolution of equilibrium points and their stability are explored. There is no collinear equilibrium points in the present model due to the effect of these parameters. All non-collinear equilibrium points are found to be unstable and the effect on positions of equilibrium points is also significant. The study of Newton–Raphson (N–R) basins of convergence and the existence of fractals in the present model is a crucial aspect of this work. Due to inclusion of variable mass parameters, the shape of basins of convergence becomes infinite in some cases. Pie charts are drawn to explain the relation between the number of iterations of N–R method and the number of initial conditions converging towards equilibrium points. Further, it is observed that the initial conditions taking less iterations of Newton–Raphson method are found away from the boundaries of basins of convergence. Due to variable mass parameters, the degree of unpredictability along boundaries of basins of convergence increases and the existence of a fractal is found in most of the cases.