# Basins of Convergence in a Multi-Perturbed CR3 BP +
## 1 Introduction  
The interest in space missions to Mars has increased significantly in recent years. Since several successful missions have reached the Martian surface [1, 2, 3], many countries and institutions have designed new missions with exploratory and scientific purposes that could pave the way for future colonization [4]. In the design of these space missions, it is essential to determine the trajectories of the spacecrafts correctly. For this purpose, the inclusion of perturbing accelerations is more and more necessary. On the other hand, optimizing the fuel consumption and resources is also very important. Placing satellites or space stations at points that could be relatively stationary with respect to nearby celestial bodies is a good solution for this optimization. Moreover, locating spacecrafts at these points can provide a strategic advantage in communication and guidance systems [5]. The study of these stationary points, known as Lagrange or libration points, is based on the \(N\)-body problem, with \(N\geq 3\). As the difficulty of finding a solution to this problem, even for \(N=3\), is significant, several simplifying hypotheses are usually incorporated with the aim of reducing the complexity of this problem in the literature. The restricted circular three-body problem (CR3 BP) [6] reduces the complexity of the general three-body problem by only considering the masses of the two primary bodies, \(m_{1}\) and \(m_{2}\), respectively, while the mass of the third body, \(m_{3}\), known as _the infinitesimal mass_, is considered negligible, so it does not alter the dynamics of the primary masses (a restricted problem). Also, the primary bodies are considered to orbit circularly (a circular problem) around their center of masses, and it is assumed that all of the bodies considered orbit on the same plane. Different modified CR3 BP models have been used for analyses of the orbital behavior of the infinitesimal mass which involve additional perturbing accelerations by modifying the potential function [7; 8; 9; 10; 11; 12; 13]. In these studies, the effects of irregularities on the shapes of the primaries, such as the radiation pressure or relativity theory, are considered to obtain more realistic models. For example, in [14], an effective potential application within the framework of multiple planets around a single star, or a binary system, can be found. However, the dynamic properties of a constrained three-body model are not limited to determining the locations of the equilibrium points but also extend to the respective basins of convergence (BCs) [15; 16]. A BC is defined as the set of initial conditions in phase space that numerically converge to a specific Lagrange point under the chosen iterative method. These attractors are determined by the gravitational and centrifugal balance in the vicinity of the equilibrium points. These dynamics tend to drive points with an infinitesimal mass and null initial velocity masses towards the Lagrange points under idealized conditions, but perturbations or transient effects can lead to divergence from these points in the short term. These regions and the libration points are calculated using iterative methods, such as Newton-Raphson's (NR) method [17] or [PERSON]'s method [18]. However, these numerical methods' convergence strongly depends on the initial conditions. Moreover, in regions near the fractal boundaries, some points may fail to converge within a finite number of iterations. These non-convergent points are excluded from the analysis of BCs. This divergence is particularly significant when the model incorporates additional perturbing forces, as these can affect both the convergence speed and the stationary behavior near the equilibrium points. The timescales and stability regions for these BCs depend on the specific parameters of the system and require further analysis in modified CR3 BP formulations.