Abstract
We establish connections between the solutions to a class of systems of ordinary differential equations of higher dimension and delay equations.
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Likhoshvaĭ V. A., Fadeev S. I., Demidenko G. V., and Matushkin Yu. G., “Modeling multistage synthesis without branching by a delay equation,” Sibirsk. Zh. Industr. Mat., 7, No. 1, 73–94 (2004).
Demidenko G. V. and Likhoshvaĭ V. A., “On differential equations with retarded argument,” Siberian Math. J., 46, No. 3, 417–430 (2005).
Demidenko G. V., Likhoshvaĭ V. A., Kotova T. V., and Khropova Yu. E., “On one class of systems of differential equations and on retarded equations,” Siberian Math. J., 47, No. 1, 45–54 (2006).
Demidenko G. V. and Mel’nik I. A., “On a method of approximation of solutions to delay differential equations,” Siberian Math. J., 51, No. 3, 419–434 (2010).
Demidenko G. V. and Kotova T. V., “Limit properties of solutions to one class of systems of differential equations with parameters,” J. Anal. Appl., 8, No. 2, 63–74 (2010).
Demidenko G. V., Kolchanov N. A., Likhoshvaĭ V. A., Matushkin Yu. G., and Fadeev S. I., “Mathematical modeling of regular contours of gene networks,” Comput. Math. Math. Phys., 44, No. 12, 2166–2183 (2004).
Murray J. D., Lectures on Nonlinear-Differential-Equation Models in Biology, Clarendon Press, Oxford (1977).
Hidirov B. N., “On one approach to modeling of living system regulatory mechanisms,” Mat. Model., 16, No. 7, 77–91 (2004).
Kotova T. V. and Mel’nik I. A., On Properties of Solutions of One Nonlinear System of Differential Equations with Parameters [in Russian] [Preprint, No. 253], Sobolev Inst. Mat., Novosibirsk (2010).
Matveeva I. I. and Mel’nik I. A., “On the properties of solutions to a class of nonlinear systems of differential equations of large dimension,” Siberian Math. J., 53, No. 2, 248–258 (2012).
Salukvadze M. E., “Concerning the synthesis of an optimal controller in linear delay systems subjected to constantly acting perturbations,” Autom. Remote Control, 23, 1495–1501 (1962).
Krasovskii N. N., “The approximation of a problem of analytic design of controls in a system with time-lag,” J. Appl. Math. Mech., 28, No. 4, 876–885 (1964).
Repin Yu. M., “On the approximate replacement of systems with lag by ordinary dynamical systems,” J. Appl. Math. Mech., 29, No. 2, 254–264 (1965).
Györi I., “Two approximation techniques for functional differential equations,” Comput. Math. Appl., 16, No. 3, 195–214 (1988).
Györi I. and Turi J., “Uniform approximation of a nonlinear delay equation on infinite intervals,” Nonlinear Anal. TMA, 17, No. 1, 21–29 (1991).
Krasznai B., Györi I., and Pituk M., “The modified chain method for a class of delay differential equations arising in neural networks,” Math. Comput. Modelling, 51, No. 5–6, 452–460 (2010).
Demidenko G. V., “On classes of systems of differential equations of a higher dimension and delay equations,” in: Itogi Nauki. Yug Rossii. Ser. Mat. Forum. Vol. 5, YuMI VNTs RAN i RSO-A, Vladikavkaz, 2011, pp. 45–56.
Demidenko G. V., “The Cauchy problem for generalized S. L. Sobolev equations,” in: Functional Analysis and Mathematical Physics [in Russian], Inst. Mat., Novosibirsk, 1985, pp. 88–105.
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Original Russian Text Copyright © 2012 Demidenko G.V.
The author was supported by the Federal Target Program “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” for 2009–2013 (State Contract 16.740.11.0127, Agreement 14.B37.21.0355), the Russian Foundation for Basic Research (Grant 10-01-00035), and Interdisciplinary Project of the Siberian Division of the Russian Academy of Sciences (Grant 80).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 6, pp. 1274–1282, November–December, 2012.
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Demidenko, G.V. Systems of differential equations of higher dimension and delay equations. Sib Math J 53, 1021–1028 (2012). https://doi.org/10.1134/S0037446612060067
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DOI: https://doi.org/10.1134/S0037446612060067