Instability of Gyroscopic Systems

A conjecture of Renshaw and Mote concerning gyroscopic systems with parameters predicts the eigenvalue locus in the neighbourhood of a double zero eigenvalue. In the present paper this conjecture is reformulated in the language of generalized eigenvectors, angular splitting and analytic behaviour of eigenvalues. Two counter-examples for systems of dimension two show that the conjecture is not generally true. Finally, splitting or analytic behaviour of eigenvalues is characterized in terms of expansion of the eigenvalues in fractional powers of the parameter.