Geodesic
Determination of the conditions of existence of limit cycles of a first-order systems with cylindrical phase space
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 1, pp. 67-76.

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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.
Keywords: phase system, limiting first-order cycles, synchronization modes, fixed point, shift operator along trajectories.
Mots-clés : quasi-synchronous modes 
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S. S. Mamonov; A. O. Harlamova. Determination of the conditions of existence of limit cycles of a first-order systems with cylindrical phase space. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 1, pp. 67-76. http://geodesic.mathdoc.fr/item/SVMO_2017_19_1_a6/

[1] V. V. Shahgildyan, A. A. Lyakhovkin, Phase-locked loop systems, Svyaz', Moscow, 1972, 446 pp. (In Russ.)

[2] G. A. Leonov, I. M. Burkin, A. I. Shepelyavyy, Frequency methods in the theory of vibrations, SPbGU, St. Petersburg, 1992, 162 pp. (In Russ.) | MR

[3] V. D. Shalfeev, V. V. Matrosov, Nonlinear dynamics of phase synchronization systems, NNGU, N. Novgorod, 2013, 366 pp. (In Russ.)

[4] S. S. Mamonov, “Sufficient conditions for the existence of limit cycles of the second kind of the frequency-phase synchronization system”, Trudy Srednevolzhskogo matematicheskogo obshchestva, 10 (2008), 203–210 (In Russ.) | Zbl

[5] S. S. Mamonov, A. O. Kharlamova, “Quasisynchronous regimes of a phase system”, Vestnik Ryazanskogo gosudarstvennogo radiotekhnicheskogo universiteta, 56 (2016), 45–51 (In Russ.)

[6] A. O. Kharlamova, “Limit cycles of the first kind of phase systems”, Vestnik RAEN. Differentsial'nye uravneniya, 3 (2016), 68–74 (In Russ.) | MR