SUMMARY There are numerous methodologies available in solving the problem in synchronizing timing oscillators of the communication networks: pleziosynchronization, forced synchronization or mutual synchronization. In the presented work the mutual synchronization system composed of four oscillators is analyzed. The mathematical model of the synchronization system is matrix differential equation with delayed argument. Applying the method of “steps” and Laplace transform we find the solution of the matrix differential equation and the step responses matrix of the synchronization system. Exact analytical and graphical expressions of transition functions and exact expressions of the phase differences between signals of oscillators of this synchronization system are obtained.

The mutual synchronization system composed of four oscillators is analysed in the work. The matrix differential equation with delayed arguments is the mathematical model of the synchronization system. The solution of the matrix differential equation is obtained applying the method of “steps” and Laplace transform. The solution involves the powers of the matrix describing the structure of internal links of the synchronization system. The powers of this matrix are calculated taking into account the eigenvalues and eigenvectors of the matrix. Applying solution of the matrix differential equation the step responses matrix of the system is obtained and transient responses are analysed. The dependence of the phase differences of the oscillators on initial conditions is analysed.

There are numerous synchronization methodologies available in solving the problem of network synchronization. This paper presents an analytical analysis of the mathematical model of forced synchronization system composed of four oscillators. Exact analytical and graphical expressions of transition functions and exact expressions of the phase differences between signals of oscillators of this synchronization system are obtained.