01-31 Heinz Hanßmann, Philip Holmes

On the global dynamics of Kirchhoff's equations : Rigid body models for underwater vehicles (1310K, PostScript, gzipped and uuencoded) Jan 19, 01

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Abstract. We study the Kirchhoff model for the motion of a rigid body submerged in an incompressible, irrotational, inviscid fluid in the absence of gravitational forces and torques. Symmetries allow reduction to a two degree-of-freedom Hamiltonian system. In [7] the existence and stability of pure and mixed mode equilibria was studied and, in [7] \S 5.2, the system was averaged, allowing further reduction to one degree of freedom. We give an interpretation of the averaged Hamiltonian function as a normal form of order one. Iterating the process we obtain the normal form of order two, thus resolving a degeneracy noted in [7], and allowing us to prove that the (integrable) normal form of order two has heteroclinic orbits between `pure 2' and between the `pure 3' modes in a range of parameter values, and, at a critical (bifurcation) value, heteroclinic cycles linking pure 2 and pure 3 modes. We discuss the implications for the original system and the full rigid body motions.

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