Analytical and Numerical Analysis of Chaos in Attitude Dynamics of a Satellite in an Elliptic Orbit

In this paper, we investigate chaos in attitude dynamics of a rigid satellite in an elliptic orbit analytically and numerically. The goal in the analytical part is to prove the existence of chaos and then to find a relation for the width of chaotic layers based on the parameters of the system. The numerical part is aimed at validating the analytical method using the Poincare maps and the plots obtained on the sensitivity to initial conditions. For this end, first, the Hamiltonian for the unperturbed system is derived. This Hamiltonian has three degrees of freedom due to the three-axis free rotation of the satellite. However, the unperturbed attitude dynamics has two first-integrals of motion, namely, the energy and the angular momentum. Next, we use the Serret-Andoyer transformation and reduce the unperturbed system Hamiltonian to one-degree of freedom. Then, the gravity gradient perturbation due to moving in an elliptic orbit is approximated in Serret-Andoyer variables and time. Due to this approximation and simplification, the system Hamiltonian transforms to a one-degree-of-freedom non-autonomous one. After that, Melnikov’s method is used to prove the existence of chaos around the heteroclinic orbits of the system. Finally, a relation for calculating the width of chaotic layers around the heteroclinic orbits in the Poincare map of the Serret-Andoyer variables is analytically derived. Results show that the analytical method gives a good approximation of the width of chaotic layers. Moreover, the results show that the analytical method is accurate even for orbits with large eccentricities.