Matrix equations of the system of phase synchronization

The system of matrix Lurie equations is considered. Such a system is of practical importance in the study of the asymptotic stability of equilibrium states of a system of differential equations, finding the regions of attraction of equilibrium states, determining the conditions for the existence of limit cycles for systems of differential equations, investigating global stability, hidden synchronization of phase and frequency-frequency self-tuning systems. It is known that the conditions for the solvability of the matrix Lurie equations are determined by the "Yakubovich–Kalman frequency theorem". To study nonlinear oscillations of phase systems, it becomes necessary to find solutions of the matrix Lurie equations.
In this paper we consider the case when the matrix Lyapunov inequality, which is part of the Lurie equation, has a matrix with real eigenvalues, some of which may be zero. For such a case, necessary and sufficient conditions for the solvability of the Lurie equations are obtained and the form of the solutions is determined, which makes it possible to carry out their spectral analysis. The explicit form of the solutions of the matrix equations made it possible to make their geometric interpretation depending on the spectrum, to show the relationship of the linear connection equation to the quadratic forms of solutions of the Lurie equations. The method of analyzing matrix equations is based on an approach based on the use of a direct product of matrices and the application of generalized inverse matrices to find solutions to systems of linear equations. The results of the work made it possible to investigate the system of three matrix equations arising in the study of phase-frequency frequency-phase self-tuning systems.