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Zbl 1015.37043
Hanssmann, Heinz; Holmes, Philip
On the global dynamics of Kirchhoff's equations: Rigid body models for underwater vehicles. (English)
[A] Broer, Henk W. (ed.) et al., Global analysis of dynamical systems. Festschrift dedicated to Floris Takens for his 60th birthday. Bristol: Institute of Physics Publishing. 353-371 (2001). ISBN 0-7503-0803-6/hbk

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 {\it P. Holmes, J. Jenkins} and {\it 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.
[Jörg Härterich (Berlin)]

MSC 2000:: *37J20 Bifurcation problems
37G05 Normal forms
37G40 Symmetries, equivariant bifurcation theory
37J15 Symmetries, etc.
70E15 Motion of rigid bodies
34C37 Homoclinic and heteroclinic solutions of ODE

Keywords: underwater vehicle; Kirchhoff's equation; averaging; normal form; heteroclinic orbits; nearly spherical body

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