Analytical and numerical study of chaos in spatial attitude dynamics of a satellite in an elliptic orbit

Abstract

In this paper, chaos in spatial attitude dynamics of a triaxial rigid satellite in an elliptic orbit is investigated 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 (i.e. the initial values needed to have a chaotic attitude motion) based on the parameters of the system. The numerical part is aimed at validating the analytical method using the Poincaré maps and the maximum value of the Lyapunov exponents. The rotational–translational Hamiltonian of the system is first derived. This Hamiltonian has six degrees of freedom. Choosing a proper set of coordinates and given the fact that the total angular momentum is constant, the Hamiltonian is then reduced to a four-degree-of-freedom system. Assuming the effect of attitude on the orbital dynamics to be negligible, and assuming a nearly symmetric and fast-spinning satellite, the system is approximated by a second-order differential equation with a time quasi-periodic perturbation. Next, the Melnikov–Wiggins’s method is used to prove the existence of a chaotic behavior followed by the determination of an analytical relation for the width of chaotic layers. Although in the analytical method some restrictive assumptions are enforced, the results show that the analytical relation gives a good estimate for the width of chaotic layers even if these assumptions are not entirely satisfied. The results also show that this method is useful for finding the effects of all the parameters (the orbit and the satellite) and the initial values on the existence of a regular behavior.