Equilibria, stability and Hamiltonian Hopf bifurcation of a gyrostat in an incompressible ideal fluid

For a gyrostat in a incompressible ideal fluid, by writing Kirchhoff’s equations as a Lie–Poisson system and using a non-canonical Hamiltonian formulation, we provide the expressions of the equilibria when the gyrostatic momentum is constant with the form l = ( 0 , 0 , l ) and present necessary and sufficient conditions for the stability of some of them via the energy–Casimir method and the study of the linearized equations of the motion. Finally, using a recent geometric method introduced by Hanssmann and Van der Meer, we give a sufficient condition for the existence of a non-degenerate Hamiltonian Hopf bifurcation at those equilibria when the gyrostat is symmetric.