We prove the existence of horseshoes in the nearly symmetric heavy top.
This problem was previously addressed but treated inappropriately due to a
singularity of the equations of motion. We introduce an (artificial) inclined
plane to remove this singularity and use a Melnikov-type approach to show
that there exist transverse homoclinic orbits to periodic orbits on
four-dimensional level sets. The price we pay for removing the singularity
is that the Hamiltonian system becomes a three-degree-of-freedom system with
an additional first integral, unlike the two-degree-of-freedom formulation in
the classical treatment. We therefore have to analyse three-dimensional
stable and unstable manifolds of periodic orbits in a six-dimensional phase
space. A new Melnikov-type technique is developed for this situation.
Numerical evidence for the existence of transverse homoclinic orbits on a
four-dimensional level set is also given.
keywords: horseshoe, heavy top, chaos, nonintegrability, Melnikov methodZ. Angew. Math. Phys. (ZAMP) 65, 221-240 (2014)