#### For citation:

Kashchenko S. A. Dynamics of full-coupled chains of a great number of oscillators with a large delay in couplings. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2023, vol. 31, iss. 4, pp. 523-542. DOI: 10.18500/0869-6632-003054, EDN: YSXPTE

# Dynamics of full-coupled chains of a great number of oscillators with a large delay in couplings

The subject of this work is the study of local dynamics of full-coupled chains of a great number of oscillators with a large delay in couplings. From a discrete model describing the dynamics of a great number of coupled oscillators, a transition has been made to a nonlinear integro-differential equation, continuously depending on time and space variable. A class of full-coupled systems has been considered. The main assumption is that the amount of delay in the couplings is large enough. This assumption opens the way to the use of special asymptotic methods of study. The parameters under which the critical case is realized in the problem of the equilibrium state stability have been distinguished. It is shown that they have infinite dimension. The analogues of normal forms — nonlinear boundary value problems of Ginzburg–Landau type have been constructed. In some cases, these boundary value problems contain integral components too. Their nonlocal dynamics describes the behavior of all solutions of the original equations in the balance state neighbourhood.

Methods. As applied to the considered problems, methods of constructing quasinormal forms on central manifolds are developed. An algorithm for constructing the asymptotics of solutions based on the use of quasinormal forms for determining slowly varying amplitudes has been created.

Results. Quasinormal forms that determine the dynamics of the original boundary value problem have been constructed. The dominant terms of asymptotic approximations for solutions of the considered chains have been obtained. On the basis of the given statements, a number of interesting dynamical effects have been revealed. For example, an infinite alternation of direct and reverse bifurcations when the delay coefficient increases. Their distinguishing feature is that they have two or three spatial variables.

- Kuznetsov AP, Kuznetsov SP, Sataev IR, Turukina LV. About Landau–Hopf scenario in a system of coupled self-oscillators. Physics Letters A. 2013;377(45–48):3291–3295. DOI: 10.1016/j.physleta. 2013.10.013.
- Osipov GV, Pikovsky AS, Rosenblum MG, Kurths J. Phase synchronization effects in a lattice of nonidentical Rossler oscillators. Phys. Rev. E. 1997;55(3):2353–2361. DOI: 10.1103/PhysRevE. 55.2353.
- Pikovsky A, Rosenblum M, Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge: Cambridge University Press; 2001. 411 p. DOI: 10.1017/CBO9780511755743.
- Dodla R, Sen A, Johnston GL. Phase-locked patterns and amplitude death in a ring of delay coupled limit cycle oscillators. Phys. Rev. E. 2004;69(5):056217. DOI: 10.1103/PhysRevE. 69.056217.
- Williams CRS, Sorrentino F, Murphy TE, Roy R. Synchronization states and multistability in a ring of periodic oscillators: Experimentally variable coupling delays. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2013;23(4):043117. DOI: 10.1063/1.4829626.
- Rao R, Lin Z, Ai X, Wu J. Synchronization of epidemic systems with Neumann boundary value under delayed impulse. Mathematics. 2022;10(12):2064. DOI: 10.3390/math10122064.
- Van der Sande G, Soriano MC, Fischer I, Mirasso CR. Dynamics, correlation scaling, and synchronization behavior in rings of delay-coupled oscillators. Phys. Rev. E. 2008;77(5):055202. DOI: 10.1103/PhysRevE.77.055202.
- Klinshov VV, Nekorkin VI. Synchronization of delay-coupled oscillator networks. Phys. Usp. 2013;56(12):1217–1229. DOI: 10.3367/UFNe.0183.201312c.1323.
- Heinrich G, Ludwig M, Qian J, Kubala B, Marquardt F. Collective dynamics in optomechanical arrays. Phys. Rev. Lett. 2011;107(4):043603. DOI: 10.1103/PhysRevLett.107.043603.
- Zhang M, Wiederhecker GS, Manipatruni S, Barnard A, McEuen P, Lipson M. Synchronization of micromechanical oscillators using light. Phys. Rev. Lett. 2012;109(23):233906. DOI: 10.1103/PhysRevLett.109.233906.
- Lee TE, Sadeghpour HR. Quantum synchronization of quantum van der Pol oscillators with trapped ions. Phys. Rev. Lett. 2013;111(23):234101. DOI: 10.1103/PhysRevLett.111.234101.
- Yanchuk S, Wolfrum M. Instabilities of stationary states in lasers with long-delay optical feedback. SIAM Journal on Applied Dynamical Systems. 2010;9(2):519–535. DOI: 10.20347/WIAS. PREPRINT.962.
- Grigorieva EV, Haken H, Kashchenko SA. Complexity near equilibrium in model of lasers with delayed optoelectronic feedback. In: 1998 International Symposium on Nonlinear Theory and its Applications (NOLTA’98). 14-17 September 1998, Crans-Montana, Switzerland. NOLTA Society; 1998. P. 495–498.
- Kashchenko SA. Quasinormal forms for chains of coupled logistic equations with delay. Mathematics. 2022;10(15):2648. DOI: 10.3390/math10152648.
- Kashchenko SA. Dynamics of a chain of logistic equations with delay and antidiffusive coupling. Doklady Mathematics. 2022;105(1):18–22. DOI: 10.1134/S1064562422010069.
- Thompson JMT, Stewart HB. Nonlinear Dynamics and Chaos. 2nd edition. New York: Wiley; 2002. 460 p.
- Kashchenko SA. Dynamics of advectively coupled Van der Pol equations chain. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2021;31(3):033147. DOI: 10.1063/5.0040689.
- Kanter I, Zigzag M, Englert A, Geissler F, Kinzel W. Synchronization of unidirectional time delay chaotic networks and the greatest common divisor. Europhysics Letters. 2011;93(6):60003. DOI: 10.1209/0295-5075/93/60003.
- Rosin DP, Rontani D, Gauthier DJ, Scholl E. Control of synchronization patterns in neural-like Boolean networks. Phys. Rev. Lett. 2013;110(10):104102. DOI: 10.1103/PhysRevLett.110.104102.
- Yanchuk S, Perlikowski P, Popovych OV, Tass PA. Variability of spatio-temporal patterns in non-homogeneous rings of spiking neurons. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2011;21(4):047511. DOI: 10.1063/1.3665200.
- Klinshov V, Nekorkin V. Synchronization in networks of pulse oscillators with time-delay coupling. Cybernetics and Physics. 2012;1(2):106–112.
- Klinshov VV. Collective dynamics of networks of active units with pulse coupling: Review. Izvestiya VUZ. Applied Nonlinear Dynamics. 2020;28(5):465–490 (in Russian). DOI: 10.18500/ 0869-6632-2020-28-5-465-490.
- Hale JK. Theory of Functional Differential Equations. 2nd edition. New York: Springer; 1977. 366 p. DOI: 10.1007/978-1-4612-9892-2.
- Hartman P. Ordinary Differential Equations. New York: Wiley; 1965. 632 p.
- Marsden JE, McCracken MF. The Hopf Bifurcation and Its Applications. New York: Springer; 1976. 408 p. DOI: 10.1007/978-1-4612-6374-6.
- Kaschenko SA. Quasinormal forms for parabolic equations with small diffusion. Soviet Mathematics. Doklady. 1988;37(2):510–513.
- Kaschenko SA. Normalization in the systems with small diffusion. International Journal of Bifurcation and Chaos. 1996;6(6):1093–1109. DOI: 10.1142/S021812749600059X.
- Kashchenko SA. The Ginzburg–Landau equation as a normal form for a second-order difference-differential equation with a large delay. Computational Mathematics and Mathematical Physics. 1998;38(3):443–451.
- Kashchenko IS, Kashchenko SA. Local dynamics of difference and difference-differential equations. Izvestiya VUZ. Applied Nonlinear Dynamics. 2014;22(1):71–92 (in Russian). 10.18500/0869-6632-2014-22-1-71-92.
- Kashchenko SA. Bifurcations in the neighborhood of a cycle under small perturbations with a large delay. Computational Mathematics and Mathematical Physics. 2000;40(5):659–668.
- Kashchenko SA. Van der Pol equation with a large feedback delay. Mathematics. 2023;11(6):1301. DOI: 10.3390/math11061301.
- Grigorieva EV, Kashchenko SA. Rectangular structures in the model of an optoelectronic oscillator with delay. Physica D: Nonlinear Phenomena. 2021;417:132818. DOI: 10.1016/j.physd.2020. 132818.
- Grigorieva EV, Kashchenko SA. Local dynamics of laser chain model with optoelectronic delayed unidirectional coupling. Izvestiya VUZ. Applied Nonlinear Dynamics. 2022;30(2):189–207. DOI: 10.18500/0869-6632-2022-30-2-189-207.
- Kashchenko SA. Quasi-normal forms in the problem of vibrations of pedestrian bridges. Doklady Mathematics. 2022;106(2):343–347. DOI: 10.1134/S1064562422050131.
- Kashchenko I, Kaschenko S. Infinite process of forward and backward bifurcations in the logistic equation with two delays. Nonlinear Phenomena in Complex Systems. 2019;22(4):407–412. DOI: 10.33581/1561-4085-2019-22-4-407-412.

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