Transversality for a class of 3D oscillators via gyrostat equations

Abstract

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.