This paper analyzes the relationship between different family branches of 2-dimensional quasi-periodic invariant tori and their linear parametrization in both the circular and elliptical restricted three-body problems. Convergence basins associated with each family branch are defined by the maximal linear parametrization magnitude \(\varepsilon \) and computed via flow map reduction under rotational invariance. Convergence basins are correlated with tori stability indices and computational failure modes at the basin edge. Sensitivities of the convergence basin with respect to the dynamical system parameters of non-dimensional mass \(\mu \) , and eccentricity e are examined. Invariant curve families expose geometric differences in tori family branches. 2-dimensional tori in the vicinity of common \(L_1\) and \(L_2\) libration point orbit families are analyzed. Regions where particular tori family branches are better represented by the linear parametrization are found, and present an important outcome of this study.
