Algorithms for the Search of Frozen Orbits in the Zonal Problem and the Identification of Ground Track Crossovers

This thesis investigates numerical methods for the search of frozen and repeating ground-track orbits in the zonal problem of artificial satellite motion, with particular attention to the identification of ground-track crossovers. Frozen orbits, characterized by long-term stability of selected orbital elements, represent an important class of solutions for Earth observation and altimetry missions, where repeatability and geometric coverage are essential requirements. The work first develops the theoretical framework of predictor–corrector techniques for conservative dynamical systems, emphasizing their application to the zonal gravitational field. In this context, the isoenergetic corrector and the tangential predictor are formulated and implemented in order to refine approximate periodic or quasi-periodic trajectories while preserving the relevant integral of motion. The numerical procedure is then extended from the reduced meridian-plane dynamics to the three-dimensional Earth-rotating reference frame, where the closure condition of the ground track is imposed. A second part of the thesis focuses on the construction of suitable initial seeds for repeating or near-repeating orbits. Building on the approaches of Farless and King, a MATLAB routine is developed to generate initial states from mission-design parameters such as the repeat cycle, inclination, and crossover geometry. These seeds are subsequently refined through the isoenergetic–tangential predictor–corrector algorithm to obtain periodic or near-periodic solutions compatible with prescribed ground-track constraints. The proposed implementation is validated through numerical experiments, including the verification of tabulated periodic solutions from the literature and the comparison with published parameters of representative satellite missions. The convergence behaviour of the algorithm is analysed by monitoring energy variations, period mismatch, correction norms, and the evolution of the associated orbital elements. The results show that the combined IC/TP strategy is able to systematically improve the periodicity of the initial seeds and to recover orbital configurations consistent with the desired repeat-ground-track structure. Overall, the thesis provides a computational framework for generating, correcting, and analysing frozen repeating ground-track orbits in the zonal problem. The developed algorithms offer a practical tool for studying periodic orbit families, assessing crossover patterns, and supporting preliminary mission-design analyses in Earth satellite dynamics.