**MSC2010** 34C25

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

#### S. S. Mamonov^{1}, A. O. Kharlamova^{2}

Annotation | In this article consider 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), the question of realizing oscillatory regimes that are associated with a violation of the stability of the equilibrium state corresponding to the synchronization regime and the formation of a stable limit cycle around this equilibrium state is little studied. Using the torus principle, the nonlocal reduction method, and the results obtained to find the solution of the system of matrix equations, the problem of developing a numerical-analytical approach for determining the conditions for the existence of limit cycles of the first kind of a system of differential equations that correspond to the oscillatory modes of the system is solved. An algorithm is developed for checking the conditions for the existence of limit cycles of the first kind, which makes it possible to determine in the phase space of the initial system a region containing initial conditions of the limiting cycle of the first kind. Applied value of the obtained results consists in the possibility of using the system (FPLL) as a generator of modulated oscillations, as well as for determining the conditions for the existence of quasisynchronous regimes of phase systems. |
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Keywords | phase system, a quasi-synchronous mode, limiting first-order cycles, timing, fixed point, the shift operator along trajectories |

^{1}**Sergei S. Mamonov**, Full Professor, Department of mathematics and methods of teaching mathematical disciplines, Ryazan State University named after S.A. Esenina (46 Svobody Str., Ryazan 390000, Russia), Dr. Sci. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-5626-748X, s.mamonov@rsu.edu.ru^{2}**Anastasiya O. Kharlamova**,Graduate Student, Department of mathematics and methods of teaching mathematical disciplines, Ryazan State University named after S.A. Esenina (46 Svobody Str., Ryazan 390000, Russia), ORCID: http://orcid.org/0000-0002-7811-381X, a.harlamova@rsu.edu.ru

**Citation**: S. S. Mamonov, A. O. Kharlamova, "[Determination of the conditions of existence of limit cycles of a first-order systems with cylindrical phase space]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:1 (2017) 67–76 (In Russian)

**DOI** 10.15507/2079-6900.19.2017.01.67-76