Abstract
Special systems of ordinary differential equations—the so called fully coupled networks of nonlinear oscillators—are considered. For this class of systems, methods for analyzing the existence and stability of solutions of the traveling wave type are proposed. A feature of the proposed methods is that they use auxiliary delay systems for finding periodic solutions and for analyzing their stability properties.
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REFERENCES
J. J. Hopfield, “Neurons with graded response have collective computational properties like those of two-state neurons,” Proc. Nat. Acad. Sci. USA. 81, 3088–3092 (1984).
S. Haykin, Neural Networks: A Comprehensive Foundation (Prentice Hall, Upper Saddle River, N.J., 1999).
V. A. Likhoshvai, Yu. G. Matushkin, and S. I. Fadeev, “Problems of the theory of gene networks,” Sib. Zh. Industr. Mat. 6 (2), 64–80 (2003).
V. A. Likhoshvai, G. V. Demidenko, S. I. Fadeev, Yu. G. Matushkin, and N. A. Kolchanov, “Mathematical simulation of regulatory circuits of gene networks,” Comput. Math. Math. Phys. 44, 2166–2182 (2004).
S. I. Fadeev and V. A. Likhoshvai, “On hypothetical gene networks,” Sib. Zh. Industr. Mat. 6 (3), 134–153 (2003).
S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “Relaxation self-oscillations in Hopfield networks with delay,” Izv. Math. 77, 271–312 (2013).
S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “On a method for mathematical modeling of chemical synapses,” Differ. Equations 49, 1193–1210 (2013).
S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “Periodic traveling-wave-type solutions in circular chains of unidirectionally coupled equations,” Theor. Math. Phys. 175, 499–517 (2013).
S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “The buffer phenomenon in ring-like chains of unidirectionally connected generators,” Izv. Math. 78, 708–743 (2014).
A. Yu. Kolesov and N. Kh. Rozov, Invariant Tori of Nonlinear Wave Equations (Fizmatlit, Moscow, 2004) [in Russian].
Funding
This work was supported by the Russian Foundation for Basic Research, project no. 18-29-10055/18.
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Translated by A. Klimontovich
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Glyzin, S.D., Kolesov, A.Y. Traveling Waves in Fully Coupled Networks of Linear Oscillators. Comput. Math. and Math. Phys. 62, 66–83 (2022). https://doi.org/10.1134/S0965542522010079
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DOI: https://doi.org/10.1134/S0965542522010079