On the global dynamics of Kirchhoff's equations: Rigid body models for underwater vehicles (Q2770239)

The motion of a rigid ellipsoidal body in an inviscid, incompressible, irrotational fluid can be modelled by Kirchhoff's equations. This leads to a Hamiltonian system with three degrees of freedom and three reversing symmetries. There exist three `pure mode' solutions in which the body moves along one of its principal axes and rotates around the same axis. In the present paper the case of a nearly spherical body is discussed. This allows to reduce the system further by averaging. It had been previously noted by \textit{P. Holmes, J. Jenkins} and \textit{N. E. Leonard} [Physica D 118, 311-342 (1998)] that the first-order normal form obtained by averaging possesses a degeneracy. This degeneracy is resolved now by computing the second-order normal form. It is shown that the normal form equations possess a range of parameters for which heteroclinic orbits between two different pure modes exist. For a critical value of the parameter there exists a heteroclinic cycle between the two pure modes. These heteroclinic orbits persist for the untruncated system and may help to design energy-efficient control mechanisms for rapid reorientation of underwater vehicles.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00062].