ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Kashchenko A. A. Influence of coupling on the dynamics of three delayed oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 6, pp. 869-891. DOI: 10.18500/0869-6632-2021-29-6-869-891

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 222)
Article type: 

Influence of coupling on the dynamics of three delayed oscillators

Kashchenko Alexandra Andreevna, P. G. Demidov Yaroslavl State University

The purpose of this study is to construct the asymptotics of the relaxation regimes of a system of differential equations with delay, which simulates three diffusion-coupled oscillators with nonlinear compactly supported delayed feedback under the assumption that the factor in front of the feedback function is large enough. Also, the purpose is to study the influence of the coupling between the oscillators on the nonlocal dynamics of the model. Methods. We construct the asymptotics of solutions of the considered model with initial conditions from a special set. From the asymptotics of the solutions, we obtain an operator of the translation along the trajectories that transforms the set of initial functions into a set of the same type. The main part of this operator is described by a finite-dimensional mapping. The study of its dynamics makes it possible to refine the asymptotics of the solutions of the original model and draw conclusions about its dynamics. Results. It follows from the form of the constructed mapping that for positive coupling parameters of the original model, starting from a certain moment of time, all three generators have the same main part of the asymptotics — the generators are “synchronized”. At negative values of the coupling parameter, both inhomogeneous relaxation cycles and irregular regimes are possible. The connection of these modes with the modes of the constructed finite-dimensional mapping is described. Conclusion. From the results of the work it follows that the dynamics of the model under consideration is fundamentally influenced by the value of the coupling parameter between the generators.

Research funded by the Council on grants of the President of the Russian Federation (MK-1028.2020.1)
  1. Ringwood JV, Malpas SC. Slow oscillations in blood pressure via a nonlinear feedback model. Am. J. Physiol. Regul. Integr. Comp. Physiol. 2001;280(4):R1105–R1115. DOI: 10.1152/ajpregu.2001.280.4.R1105.
  2. Glyzin SD, Kolesov AY, Rozov NK. Buffering in cyclic gene networks. Theor. Math. Phys. 2016;187(3):935–951. DOI: 10.1134/S0040577916060106.
  3. Karavaev AS, Ishbulatov YM, Ponomarenko VI, Prokhorov MD, Gridnev VI, Bezruchko BP, Kiselev AR. Model of human cardiovascular system with a loop of autonomic regulation of the mean arterial pressure. J. Am. Soc. Hypertens. 2016;10(3):235–243. DOI: 10.1016/j.jash.2015.12.014.
  4. Karavaev AS, Ishbulatov YM, Prokhorov MD, Ponomarenko VI, Kiselev AR, Runnova AE, Hramkov AN, Semyachkina-Glushkovskaya OV, Kurths J, Penzel T. Simulating dynamics of circulation in the awake state and different stages of sleep using non-autonomous mathematical model with time delay. Front. Physiol. 2021;11:612787. DOI: 10.3389/fphys.2020.612787.
  5. an der Heiden U, Mackey MC. The dynamics of production and destruction: Analytic insight into complex behavior. J. Math. Biol. 1982;16(1):75–101. DOI: 10.1007/BF00275162.
  6. Erneux T. Applied Delay Differential Equations. New York: Springer-Verlag; 2009. 204 p. DOI: 10.1007/978-0-387-74372-1.
  7. Dmitriev AS, Kislov VY. Stochastic Oscillations in Radiophysics and Electronics. Moscow: Nauka; 1989. 280 p. (in Russian).
  8. Kilias T, Kelber K, Mogel A, Schwarz W. Electronic chaos generators – design and applications. International Journal of Electronics. 1995;79(6):737–753. DOI: 10.1080/00207219508926308.
  9. Kashchenko SA, Maiorov VV. Models of Wave Memory. Moscow: Book House «Librokom»; 2009. 288 p. (in Russian).
  10. Ponomarenko VI, Prokhorov MD, Karavaev AS, Kulminskiy DD. An experimental digital communication scheme based on chaotic time-delay system. Nonlinear Dyn. 2013;74(4):1013–1020. DOI: 10.1007/s11071-013-1019-0.
  11. Lakshmanan M, Senthilkumar DV. Dynamics of Nonlinear Time-Delay Systems. Berlin, Heidelberg: Springer-Verlag; 2011. 313 p. DOI: 10.1007/978-3-642-14938-2.
  12. Preobrazhenskaya MM. Discrete traveling waves in a relay system of Mackey–Glass equations with two delays. Theor. Math. Phys. 2021;207(3):827–840. DOI: 10.1134/S0040577921060106.
  13. Mallet-Paret J, Nussbaum RD. Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation. Annali di Matematica Pura ed Applicata. 1986;145(1): 33–128. DOI: 10.1007/BF01790539.
  14. Losson J, Mackey MC, Longtin A. Solution multistability in first-order nonlinear differential delay equations. Chaos. 1993;3(2):167–176. DOI: 10.1063/1.165982. 
  15. Krisztin T, Walther HO. Unique periodic orbits for delayed positive feedback and the global attractor. J. Dyn. Diff. Equat. 2001;13(1):1–57. DOI: 10.1023/A:1009091930589.
  16. Stoffer D. Delay equations with rapidly oscillating stable periodic solutions. J. Dyn. Diff. Equat. 2008;20(1):201–238. DOI: 10.1007/s10884-006-9068-4.
  17. Krisztin T, Vas G. Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback. J. Dyn. Diff. Equat. 2011;23(4):727–790. DOI: 10.1007/s10884-011-9225-2.
  18. Kashchenko I, Kaschenko S. Normal and quasinormal forms for systems of difference and differential-difference equations. Communications in Nonlinear Science and Numerical Simulation. 2016;38:243–256. DOI: 10.1016/j.cnsns.2016.02.041.
  19. Kashchenko AA. Non-rough relaxation solutions of a system with delay and sign-changing nonlinearity. Nonlinear Phenomena in Complex Systems. 2019;22(2):190–195.
  20. Kashchenko AA. Relaxation cycles in a model of two weakly coupled oscillators with signchanging delayed feedback. Theor. Math. Phys. 2020;202(3):381–389. DOI: 10.1134/S0040577920030101.
  21. Kashchenko AA. Dependence of dynamics of a system of two coupled generators with delayed feedback on the sign of coupling. Mathematics. 2020;8(10):1790. DOI: 10.3390/math8101790.
  22. Kashchenko AA. Relaxation modes of a system of diffusion coupled oscillators with delay. Communications in Nonlinear Science and Numerical Simulation. 2021;93:105488. DOI: 10.1016/j.cnsns.2020.105488.
  23. Schauder J. Der Fixpunktsatz in Funktionalraumen. Studia Mathematica. 1930;2(1):171–180 (in German). DOI: 10.4064/sm-2-1-171-180.