Stability sets of multiparameter Hamiltonian systems

A real linear Hamiltonian system with constant coefficients that depend on several real parameters is considered. A method is proposed for calculating the sets of all values of the parameters for which the stationary solution of this system is stable for fixed values of the parameters (that is, the stability sets). The application of the method is demonstrated for a gyroscopic problem described by a Hamiltonian system with four degrees of freedom and three parameters. Computer algebra, in particular, a Gröbner basis and a Power Geometry are used. It is shown that the four-parameter generalization of this problem does not contain fundamentally new difficulties.