# Basins of Convergence in a Multi-Perturbed CR3 BP +
## 3 Lagrange Points and Basins of Convergence  
This section describes the Lagrange points of the system and the calculation methodology implemented to obtain them, which involves the use of the NR method. As shown in Section 2, the CR3 BP is based on defining a potential function whose value depends on the coordinates of the infinitesimal mass. Given the definition of this effective potential, it is possible to find points where this function presents maximums and/or minimums. These points are known as Lagrange points or libration points and determine the places in space where an infinitesimal mass, affected only by the gravitational field exerted by the two primary masses, can reach a stationary position with respect to the latter. At these points, the position of the third mass can remain stationary since the gravitational field exerted by the primary bodies is compensated for by the centrifugal acceleration of the infinitesimal mass.  
In general, there are a total of five libration points, denoted as \(L_{1}\), \(L_{2}\), \(L_{3}\), \(L_{4}\), and \(L_{5}\). The first three are known as collinear points and are located on axis \(x\) (\(y=0\)), that is, on the line joining the two primary masses. On the other hand, points \(L_{4}\) and \(L_{5}\), known as triangular points, are points that lie outside the mentioned axis, so their coordinate \(y\) is not zero.  
Analogously, when considering natural satellites and other effects, a modified CR3 BP results in a new effective potential function (see Section 2). The introduction of new terms into the dynamic model does not affect the definition of the Lagrange points, but now, all of the added phenomena are considered in addition to the gravitational field. Thus, more than five libration points can appear. Some authors have shown how additional libration points appear when modifying the dynamic model of the CR3 BP with the effect of the oblateness of the primary masses [21].  
As is known, the libration points act as attractors within the gravitational field exerted by the two primary masses (including any modifications deemed appropriate). That is why an infinitesimal mass located at \((x_{0},y_{0})\) will naturally tend to approach one of the aforementioned points. The basins of convergence determine which Lagrange point an infinitesimal mass evolves towards given its initial conditions \((x_{0},y_{0})\) and assuming that its initial velocity is zero \((\dot{x}_{0}=\dot{y}_{0}=0)\). While these regions mathematically define the dynamic attractors for specific initial conditions, real objects may not remain stable at the Lagrange points due to perturbations or orbital eccentricities. Instead, they may exhibit periodic or quasi-periodic motion in their vicinity.  
Taking into account the fact that the condition that must be met for a point in space to be a libration point is for the potential to reach the maximum or the minimum, the equations that define them are the following:  
\[\begin{array}{l}\Omega^{*}{}_{x}=0\\ \Omega^{*}{}_{y}=0\end{array} \tag{18}\]  
In order to solve these equations and determine the convergence basins, a certain portion of the plane is taken and meshed, dividing it into \(n_{x}\times n_{y}\) uniformly distributed elements. The centroid of each of these cells represents position \((x_{0},y_{0})\). Then, two different iterative methods are used to find the libration point towards which an infinitesimal mass without velocity and located at \((x_{0},y_{0})\) will tend. So, Equation (18) is solved by means of a searching algorithm, taking as the centroid of each cell as the initial solution.