Determination of the conditions of existence of limit cycles of a first-order systems with cylindrical phase space

This article deals with a system of differential equations with a cylindrical phase space, which is a mathematical model of a frequency-phase locked loop (FPLL) system. For the system (FPLL) little studied is the implementation of oscillatory regimes that are associated with a violation of stability of the equilibrium state corresponding to the synchronization regime and to the formation of stable limit cycle around this equilibrium state. A numerical-analytical approach is developed to determine the conditions of existence of the first kind limiting cycles of a differential equations’ system that correspond to the system oscillatory modes. To do this, author used the torus principle, the nonlocal reduction method and the results obtained to find the solution of the system of matrix equations. An algorithm is developed for checking the conditions for the existence of limit cycles of the first kind, which allows to determine a region in the phase space of initial system that contains containing initial conditions of the cycle. Applicative value of the obtained results is in the possibility of using the system (FPLL) for generation of modulated oscillations, as well as for determining the conditions of the existence of phase systems’ quasisynchronous regimes.