Chaotic attitude motion of gyrostat satellite via Melnikov method

In this paper Deprit 's variables are used to describe the Hamiltonian equations for attitude motions of a gyrostat satellite spinning about arbitrarily body-fixed axes. The Hamiltonian equations for the attitude motions of the gyrostat satellite in terms of the Deprit ' ds variables and under small viscous damping and nonautonomous perturbations are suitable for the employment of the Melnikov 's integral. The torque-free homoclinic orbits to the symmetric Kelvin gyrostat are derived by means of the elliptic function integral theory. With the help of residue theory of complex functions, the Melnikov 's integral is utilized to analytically study the criterion for chaotic oscillations of the attitude motions of the symmetric Kelvin gyrostat under small, damping and periodic external disturbing torques. The Melnikov 's integral yields an analytical criterion for the chaotic oscillations of the attitude motions in the form of an inequality that gives a necessary condition for chaotic dynamics in terms of the physical parameters. The dependence of the onset of homoclinic orbits on quantities such as body shapes, the initial conditions of the angular velocities or the two constants of motions of the torque-free gyrostat satellite is investigated in details. The dependence of the onset of chaos on quantities such as the amplitudes of the external excitation and the damping coefficients ' matrix is discussed. The bifurcation curves based upon the Melnikov 's integral are computed by using the combined parameters versus the frequency of the external excitation. The theoretical criterion agrees with the result of the numerical simulation of the gyrostat satellite by using the fourth-order Runge-Kutta integration algorithm. The numerical solutions show that the motions of the perturbed symmetric gyrostat satellite possess a lot of "random" characteristic associated with a nonperiodic solution.