Chaotic attitude motion of gyrostat satellite via Melnikov method

In this paper Deprit 's variables are used to describe the Hamiltonian equa tions for attitude motions of a gyrostat satellite spinning about arbitrari ly body-fixed axes. The Hamiltonian equations for the attitude motions of t he gyrostat satellite in terms of the Deprit ' ds variables and under small viscous damping and nonautonomous perturbations are suitable for the emplo yment of the Melnikov 's integral. The torque-free homoclinic orbits to the symmetric Kelvin gyrostat are derived by means of the elliptic function in tegral theory. With the help of residue theory of complex functions, the Me lnikov 's integral is utilized to analytically study the criterion for chao tic oscillations of the attitude motions of the symmetric Kelvin gyrostat u nder 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 co ndition for chaotic dynamics in terms of the physical parameters. The depen dence of the onset of homoclinic orbits on quantities such as body shapes, the initial conditions of the angular velocities or the two constants of mo tions of the torque-free gyrostat satellite is investigated in details. The dependence of the onset of chaos on quantities such as the amplitudes of t he 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 excitati on. The theoretical criterion agrees with the result of the numerical simul ation of the gyrostat satellite by using the fourth-order Runge-Kutta integ ration algorithm. The numerical solutions show that the motions of the pert urbed symmetric gyrostat satellite possess a lot of "random" characteristic associated with a nonperiodic solution.