  The class of 3D oscillators of interest includes the modified Brockett, Chua,
Duffing, Ueda, modified Kapitaniak, generalized Lorenz, forced Lorenz, Rossler
and YSVO oscillators. The homoclinic orbits of a symmetric gyrostat with wheels
under torque-free motions are first exploited. The effects of the small
external perturbation torques upon the rotational motions of the forced
symmetric gyrostat are investigated using the equation of the Melnikov
integral. The real zeros of the Melnikov integral determine the transversality
of the homoclinic orbits leading to a necessary condition for the existence of
chaos. The equations of the 3D oscillators are then reduced to the Euler
equations of the perturbed rotational motions of symmetric gyrostats.
Algorithms are established to compute the required parameters for the gyrostat
to represent the 3D oscillators at initiating the transversality. These
parameters include the angular momenta of the wheels and the principal moments
of inertia. The existence of real zeros of the Melnikov integral for the
symmetric gyrostat implies the existence of transversal intersections of the
perturbed solutions of the 3D oscillators. The 4th order Runge-Kutta algorithm
is utilized to simulate and crosscheck the long-term chaotic behaviors of the
dynamical systems.
