DOI 10.15507/2079-6900.23.202101.28–42

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 34C25

Dynamics of the mathematical model of phase-locked systems with delay

S. S. Mamonov, I. V. Ionova, A. O. Kharlamova

Ryazan State University named after S.A. Esenin (Ryazan, Russian Federation)

Abstract. In the article, the conditions for the existence of limit cycles of the first kind are obtained for self-tuning systems with delay, which, in turn, determine the conditions for the occurrence of hidden synchronization modes in such systems. The principle of the proof is based on constructing a positively invariant toroidal set using two cylindrical surfaces, whose boundaries are determined by the limit cycles of a system of the second-order differential equations. Using the results obtained in the article for limit cycles, the possibility of using the curvature of the cycle for a comparative analysis of the proximity of the cycles of phase and non-phase systems, as well as for determining the mode of hidden synchronization, is shown.

Key Words: system of differential equations, phase system, limit cycles of the first kind, latent synchronization, multistability, fixed point, shift operator, rotation of a vector field, cycle curvature

For citation: S. S. Mamonov, I. V. Ionova, A. O. Kharlamova. Dynamics of the mathematical model of phase-locked systems with delay. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 23:1(2021), 28–42. DOI: https://doi.org/10.15507/2079-6900.23.202101.28–42

Submitted: 22.01.2021; Revised: 25.02.2021; Accepted: 28.02.2021

Information about the authors:

Sergei S. Mamonov, Full Professor, Department of mathematics and methods of teaching mathematical disciplines, Ryazan State University named after S.A. Esenin (46 Svobody Str., Ryazan 390000, Russia), D. Sci. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-5626-748X, s.mamonov@365.rsu.edu.ru

Irina V. Ionova, Associate Professor, Department of mathematics and methods of teaching mathematical disciplines, Ryazan State University named after S.A. Esenin (46 Svobody Str., Ryazan 390000, Russia), ORCID: https://orcid.org/0000-0002-2580-5388, i.ionova@365.rsu.edu.ru

Anastasiya O. Kharlamova, Senior Lecturer, Department of mathematics and methods of teaching mathematical disciplines, Ryazan State University named after S.A. Esenin (46 Svobody Str., Ryazan 390000, Russia), Ph. D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-7811-381X, a.harlamova@365.rsu.edu.ru

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.