Homoclinic orbits of the Kovalevskaya top with perturbations

Abstract

In this paper the instability issue of the permanent rotation of a heavy top is revisited and the analytical characteristic equation for the particular solution is derived. The homoclinic orbits of the Kovalevskaya top are formulated from the Kovalevskaya fundamental equation and the Kotter transformation. Some integrable motions of the undisturbed Kovalevskaya top are obtained by means of the Jacobian elliptic integrals. The criteria for judging the onset of homoclinic transversal intersections of the stable and unstable manifolds at a saddle in the Poincaré map when the Kovalevskaya top is disturbed by a small external torque are established via the Melnikov integral due to Holmes and Marsden [15]. This theoretical achievement is crosschecked by the 4th-order Runge-Kutta algorithms and by the Poincaré section to investigate the long-term behaviors of the Euler-Poisson equations with small forced torques. This also gives a theoretical and numerical evidence for the nonintegrability of the disturbed Kovalevskaya top.