A mechanical system with a ‘‘wild’’ horseshoe

Abstract

We consider a gyrostat in its simplest form, namely, a rigid body with a flywheel attached. It is well known that, although the motions are coupled, when the rotor is symmetric, the full system is completely integrable in terms of elliptic functions. The Euler equations for the angular momentum components with respect to the rigid body now depend on the parameter I=angular momentum of the flywheel (a constant of motion) and the phase portrait of the system undergoes several bifurcations. Analytical formulas, in terms of elementary functions, can be given for the separatrices. A small imperfection in the flywheel breaks the S1 symmetry; as a result, the separatrices split with transversal intersection, producing horseshoes. We applied the techniques of Holmes and Marsden to write down the Melnikov function of the system. The integral is computed by the method of residues, in the limit case I→0−. Somewhat surprisingly, the amplitude of the Melnikov function diverges. We propose an explanation for this ‘‘paradox.’’