Comments on stability properties of conservative gyroscopic systems

A conjecture of Renshaw and Mote concerning gyroscopic systems with paramet ers predicts the eigenvalue locus in the neighborhood of a double-zero eige nvalue. In the present paper, this conjecture is reformulated ba the langua ge of generalized eigenvectors, angular splitting, and analytic behavior of eigenvalues Two counter-examples for systems of dimension two show that th e conjecture is not generally true. Finally, splitting or analytic behavior of eigenvalues is characterized in terms of expansion of the eigenvalues i n fractional powers of the parameter.