Synchronization in a ring of oscillators with delayed feedback. (English. Russian original) Zbl 07885094
Math. Notes 115, No. 6, 944-958 (2024); translation from Mat. Zametki 115, No. 6, 879-896 (2024).
Summary: A ring of coupled oscillators with delayed feedback with various types of coupling between the oscillators is considered. For each type of coupling, the asymptotic behavior of the model solutions with respect to a large parameter is constructed for a wide variety of initial conditions. It is shown that the studying the behavior of solutions to the original infinite-dimensional models can be reduced to studying the dynamics of the constructed finite-dimensional mappings. High quality conclusions about the dynamics of the original systems are made. It is shown that the behavior of solutions significantly varies with variations in the type of coupling. Conditions on the system parameters are found under which the synchronization, two-cluster synchronization, and more complex modes are possible.
MSC:
| 34K24 | Synchronization of functional-differential equations |
Keywords:
delay; relaxation oscillations; coupled oscillators; synchronization; nonlinear systems; multidimensional systemsReferences:
| [1] | Heiden, U. an der; Mackey, M. C., The dynamics of production and destruction: analytic insight into complex behavior, J. Math. Biol., 16, 1, 75-101, 1982 · Zbl 0523.93038 · doi:10.1007/BF00275162 |
| [2] | Kislov, V. Ya.; Dmitriev, A. S., Nonlinear stochastization of oscillations in radio engineering and electronic systems, Problems of Modern Radio Engineering and Electronics, 1987, Moscow: Naika, Moscow |
| [3] | Kilias, T.; Kelber, K.; Mogel, A.; Schwarz, W., Electronic chaos generators—design and applications, Inter. J. Electronics, 79, 6, 737-753, 1995 · doi:10.1080/00207219508926308 |
| [4] | Erneux, T., Applied Delay Differential Equations, 2009, New York: Springer, New York · Zbl 1201.34002 · doi:10.1007/978-0-387-74372-1_4 |
| [5] | Lakshmanan, M.; Senthilkumar, D. V., Dynamics of Nonlinear Time-Delay Systems, 2011, Heidelberg: Springer, Heidelberg · Zbl 1230.37001 · doi:10.1007/978-3-642-14938-2 |
| [6] | Smith, H., An Introduction to Delay Differential Equations with Applications to the Life Sciences, 2011, New York: Springer, New York · Zbl 1227.34001 |
| [7] | Wu, J., Introduction to Neural Dynamics and Signal Transmission Delay, 2011, Berlin: de Gruyter, Berlin |
| [8] | Ivanov, A. F.; Sharkovsky, A. N., Oscillations in singularly perturbed delay equations, Dynamics Reported: Expositions in Dynamical Systems, 164-224, 1992, Berlin: Springer, Berlin · Zbl 0755.34065 · doi:10.1007/978-3-642-61243-5_5 |
| [9] | Krisztin, T.; Walther, H.-O., Unique periodic orbits for delayed positive feedback and the global attractor, J. Dyn. Differ. Equations, 13, 1, 1-57, 2001 · Zbl 1008.34061 · doi:10.1023/A:1009091930589 |
| [10] | Stoffer, D., Delay equations with rapidly oscillating stable periodic solutions, J. Dyn. Differ. Equations, 20, 1, 201-238, 2008 · Zbl 1204.34090 · doi:10.1007/s10884-006-9068-4 |
| [11] | Krisztin, T., Global dynamics of delay differential equations, Period. Math. Hungar., 56, 1, 83-95, 2008 · Zbl 1164.34037 · doi:10.1007/s10998-008-5083-x |
| [12] | Kashchenko, S. A., Asymptotic behavior of relaxation oscillations in systems of difference-differential equations with a compactly supported nonlinearity. I., Differ. Equations, 31, 8, 1275-1285, 1995 · Zbl 0863.34076 |
| [13] | Kashchenko, A. A., Dependence of dynamics of a system of two coupled generators with delayed feedback on the sign of coupling, Mathematics, 8, 10, 1790, 2020 · doi:10.3390/math8101790 |
| [14] | Kashchenko, A. A., Influence of coupling on the dynamics of three delayed oscillators, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelinein. Dinam., 29, 6, 869-891, 2021 |
| [15] | Kashchenko, A. A., Dependence of the dynamics of a model of coupled oscillators on the number of oscillators, Dokl. Math., 104, 3, 355-359, 2021 · Zbl 1539.34080 · doi:10.1134/S1064562421060090 |
| [16] | Kashchenko, A.; Kashchenko, I.; Kondratiev, S., Travelling waves in the ring of coupled oscillators with delayed feedback, Mathematics, 11, 13, 2827, 2023 · doi:10.3390/math11132827 |
| [17] | Glyzin, S. D.; Kolesov, A. Yu.; Rozov, N. Kh., Periodic traveling-wave-type solutions in circular chains of unidirectionally coupled equations, Theor. Math. Phys., 175, 1, 499-517, 2013 · Zbl 1364.34058 · doi:10.1007/s11232-013-0041-1 |
| [18] | Pikovsky, A.; Rosenblum, M.; Kurths, J., Synchronization. A Universal Concept in Nonlinear Sciences, 2003, Cambridge: Cambridge Univ. Press, Cambridge · Zbl 1219.37002 |
| [19] | Strogatz, S. H., Sync. How Order Emerges from Chaos in the Universe, Nature, and Daily Life, 2012, New York: Hyperion Books, New York |
| [20] | Winfree, A. T., The Geometry of Biological Time, 1980, Berlin-New York: Springer, Berlin-New York · Zbl 0464.92001 · doi:10.1007/978-3-662-22492-2 |
| [21] | Klinshov, V. V.; Zlobin, A. A., Kuramoto Model with Delay: The Role of the Frequency Distribution, Mathematics, 11, 10, 2325, 2023 · doi:10.3390/math11102325 |
| [22] | Hale, J.; Lunel, S. M. V., Introduction to Functional-Differential Equations, 1993, New York: Springer-Verlag, New York · Zbl 0787.34002 · doi:10.1007/978-1-4612-4342-7 |
| [23] | Lusternik, L.; Sobolev, V., Elements of Functional Analysis, 1968, New York: Gordon and Breach, New York |
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