The Holomorphic Regularization Method of the Tikhonov System of Differential Equations for Mathematical Modeling of Wave Solid-State Gyroscope Dynamics

This paper develops the holomorphic regularization method of the Cauchy problem for a special type of Tikhonov system that arises in the mathematical modeling of wave solid- state gyroscope dynamics. The Tikhonov system is a system of differential equations a part of which is singularly perturbed. Unlike other asymptotic methods giving approximations in the form of asymptotically converging series, the holomorphic regularization method allows one to obtain solutions of nonlinear singularly perturbed problems in the form of series in powers of a small parameter converging in the usual sense. Also, as a result of applying the holomorphic regularization method, merged formulas for an approximate solution are deduced both in the boundary layer and outside it. These formulas allow a qualitative analysis of the approximate solution on the entire time interval including the boundary layer.
This paper consists of two sections. In Section 1, the holomorphic regularization method of the Cauchy problem for a special type of Tikhonov system is developed. The special type of Tikhonov system means the following: singularly perturbed equations are linear in the variables included in them with derivatives, the matrix of the singularly perturbed part of the system is diagonal, the remaining equations have separate linear and nonlinear parts. An algorithm for deriving an approximate solution to the Cauchy problem for the Tikhonov system of special type by using the holomorphic regularization method is presented. In Section 2, the mathematical model describing in interconnected form the mechanical oscillations of the gyroscope resonator and the electrical processes in the oscillation control circuit is considered. The algorithm for deriving an approximate solution proposed in Section 1 is used. Formulas for an approximate solution taking into account the structure of the Tikhonov system are deduced.