Employing the Poincaré mapping method to identify the periodic Lyapunov orbits in the restricted three-body problem considering primaries’ oblateness

The study was done to identify periodic Lyapunov orbits in the presence of the primaries oblateness applying Poincaré map at the restricted three-body. Governing equations of the PRTBP orbital motion were derived using principles of the Lagrangian mechanics. Since the governing equations have no closed-form solution, so the numerical method must be applied. So the problem can have different periodic or non-periodic responses to the initial conditions. The proper initial conditions were obtained from combining the third-order approximation of the Unperturbed Restricted Three-Body Problem and the orbital correction algorithm in previous researches. This method required complex and time-consuming mathematical calculations. Therefore, in this paper, the suitable initial conditions of periodic Lyapunov orbit are suggested to identify with the Poincaré map. Poincaré maps are a valuable tool for capturing the dynamical structures of a system, such as periodic solutions via a discrete and lower-dimensional representation of the dynamical flow. The center and boundaries of the islands created in this map are considered as suitable initial conditions to meet the periodic responses. To validate the proposed method, the perturbed Lyapunov orbits family is plotted. Also, in order to illustrate the effect of perturbations, the initial conditions of the perturbed and unperturbed models are compared due to the same initial guess vectors.