Piecewise constant modes of operation of fully coupled networks and their limit integro-differential systems
https://doi.org/10.18255/1818-1015-2025-2-206-224
Abstract
Fully connected networks of oscillators and their limit systems of integro-differential equations with periodic boundary conditions are considered. It is assumed that the connection is weak, i.e. the coefficient at the integral term is small. In the problem of stability of the zero equilibrium state, the simplest critical cases of loss of stability are distinguished. In these situations, quasi-normal forms are constructed, which are integro-differential equations for which several continuous families of piecewise constant two-step solutions are analytically determined, and their stability is studied. The existence of piecewise constant solutions with more than one discontinuity point is shown. A numerical experiment illustrating the analytical constructions is performed.
About the Authors
Sergey D. Glyzin
P.G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
Sergey A. Kashchenko
P.G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
Dmitry S. Kosterin
P.G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
References
1. S. A. Kashchenko, “Dynamics of full-coupled chains of a great number of oscillators with a large delay in couplings,” Izvestiya VUZ. Applied Nonlinear Dynamics, vol. 31, no. 4, pp. 523–542, 2023, doi: 10.18500/0869-6632-003054.
2. D. S. Glyzin, S. D. Glyzin, and A. Y. Kolesov, “Hunt for chimeras in fully coupled networks of nonlinear oscillators,” Izvestiya VUZ. Applied Nonlinear Dynamics, vol. 30, no. 2, pp. 152–175, 2022, doi: 10.18500/0869-6632-2022-30-2-152-175.
3. S. A. Kashchenko, “Quasinormal Forms for Chains of Coupled Logistic Equations with Delay,” Mathematics, vol. 10, no. 15, p. 2648, 2022, doi: 10.3390/math10152648.
4. S. A. Kashchenko, “Asymptotics of Regular and Irregular Solutions in Chains of Coupled van der Pol Equations,” Mathematics, vol. 11, no. 9, p. 2047, 2023, doi: 10.3390/math11092047.
5. S. D. Glyzin and A. Y. Kolesov, “Periodic two-cluster synchronization modes in fully coupled networks of nonlinear oscillators,” Teoreticheskaya i Matematicheskaya Fizika, vol. 212, no. 2, pp. 213–233, 2022, doi: 10.4213/tmf10191.
6. P. Hartman, Ordinary differential equations. Boston: Birkh"auser Baltimore, 1973.
7. J. L. Dalecki and M. G. Krein, Stability of Solutions of Differential Equation in Banach Space. AMS, Providence, 1974.
8. A. D. Myshkis, “Mixed Functional Differential Equations,” Journal of Mathematical Sciences, vol. 129, pp. 4111–4226, 2005, doi: 10.1007/s10958-005-0345-2.
9. E. V. Grigorieva and S. A. Kashchenko, “Antiphase and in-phase dynamics in laser chain models with delayed bidirectional couplings,” Physica D: Nonlinear Phenomena, vol. 467, p. 134223, 2024, doi: 10.1016/j.physd.2024.134223.
10. E. V. Grigorieva and S. A. Kashchenko, “Rectangular structures in the model of an optoelectronic oscillator with delay,” Physica D: Nonlinear Phenomena, vol. 417, p. 132818, 2021, doi: 10.1016/j.physd.2020.132818.
11. J. Medlock and M. Kot, “Spreading disease: integro-differential equations old and new,” Mathematical Biosciences, vol. 184, no. 2, pp. 201–222, 2003, doi: 10.1016/S0025-5564(03)00041-5.
12. O. Bonnefon, J. Coville, J. Garnier, and L. Roques, “Inside dynamics of solutions of integro-differential equations,” Discrete and Continuous Dynamical Systems -- B, vol. 19, no. 10, pp. 3057–3085, 2014, doi: 10.3934/dcdsb.2014.19.3057.
13. Y. Jin and X.-Q. Zhao, “Spatial dynamics of a periodic population model with dispersal,” Nonlinearity, vol. 22, no. 5, p. 1167, 2009, doi: 10.1088/0951-7715/22/5/011.
14. J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications. Applied Mathematical Sciences, volume 19. Springer-Verlag, 1976.
15. B. D. Hassard, N. D. Kazarinoff, and Y.-H. Wan, Theory and applications of Hopf bifurcation. Cambridge; New York : Cambridge University Press, 1981.
16. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed. New York: Springer-Verlag, 2004.
17. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics. Boston: Academic Press, 1993.
18. D. Henry, Geometric Theory of Semilinear Parabolic Equations. Springer, 1981.
19. J. Hale, Theory of functional differential equations. New-York : Springer-Verlag, 1971.
20. J. Wu, Theory and applications of partial functional-differential equations. New York: Springer-Verlag, 1996.
21. S. A. Kashchenko, “Quasinormal forms for parabolic equations with small diffusion,” Soviet Mathematics. Doklady, vol. 37, no. 5, pp. 510–513, 1988.
22. S. A. Kashchenko, “Normalization in the systems with small diffusion,” International Journal of Bifurcation and Chaos, vol. 6, no. 6, pp. 1093–1109, 1996, doi: 10.1142/S021812749600059X.
Review
For citations:
Glyzin S.D., Kashchenko S.A., Kosterin D.S. Piecewise constant modes of operation of fully coupled networks and their limit integro-differential systems. Modeling and Analysis of Information Systems. 2025;32(2):206-224. (In Russ.) https://doi.org/10.18255/1818-1015-2025-2-206-224
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