Abstract:
This paper presents the stability of resonant rotation of a symmetric gyrostat under
third- and fourth-order resonances, whose center of mass moves in an elliptic orbit in a central
Newtonian gravitational field. The resonant rotation is a special planar periodic motion of
the gyrostat about its center of mass, i. e., the body performs one rotation in absolute space
during two orbital revolutions of its center of mass. The equations of motion of the gyrostat
are derived as a periodic Hamiltonian system with three degrees of freedom and a constructive
algorithm based on a symplectic map is used to calculate the coefficients of the normalized
Hamiltonian. By analyzing the Floquet multipliers of the linearized equations of perturbed
motion, the unstable region of the resonant rotation and the region of stability in the first-order
approximation are determined in the dimensionless parameter plane. In addition, the third-
and fourth-order resonances are obtained in the linear stability region and further nonlinear
stability analysis is performed in the third- and fourth-order resonant cases.