On a method of studying certain problems in perturbation theory

A new algebraic method for solving certain perturbation and bifurcation problems is presented. In contrast to known classical results, the algorithm proposed here allows one to solve the indicated problems more simply and constructively in a rather general situation, including a multidimensional perturbation, and also in the presence of multiple points in the spectrum of the limit operator.
In bifurcation theory the indicated method offers the possibility of finding the solution analytically with required accuracy, allowing one to bypass the traditional (in essence graphic) method of Newton diagrams.
In the theory of gyroscopes a simpler and at the same time more rigorous method for determining oscillation frequencies of an annular nonideal resonator is found with the aid of the new approach.