maisМоделирование и анализ информационных системModeling and Analysis of Information Systems1818-10152313-5417Yaroslavl State University10.18255/1818-1015-2025-2-206-224mais-1940Research ArticleComputing Methodologies and ApplicationsКусочно-постоянные режимы работы полносвязных сетей и их предельных интегро-дифференциальных системPiecewise constant modes of operation of fully coupled networks and their limit integro-differential systemshttps://orcid.org/0000-0002-6403-4061ГлызинСергей ДмитриевичGlyzinSergey D.glyzin@uniyar.ac.ruhttps://orcid.org/0000-0002-4846-6040КащенкоСергей АлександровичKashchenkoSergey A.kasch@uniyar.ac.ruhttps://orcid.org/0009-0003-6385-8063КостеринДмитрий СергеевичKosterinDmitry S.kosterin.dim@mail.ruЯрославский государственный университет им.П.Г. ДемидоваРоссияP.G. Demidov Yaroslavl State UniversityRussian Federation202528062025322206224Copyright © Глызин С.Д., Кащенко С.А., Костерин Д.С., 20252025Глызин С.Д., Кащенко С.А., Костерин Д.С.Glyzin S.D., Kashchenko S.A., Kosterin D.S.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.mais-journal.ru/jour/article/view/1940

Рассматриваются полносвязные сети осцилляторов и их предельные системы интегро-дифференциальных уравнений с периодическими краевыми условиями. Предполагается, что связь слабая, то есть мал коэффициент при интегральном члене. В задаче об устойчивости нулевого состояния равновесия выделяются простейшие критические случаи потери устойчивости. В этих ситуациях строятся квазинормальные формы, представляющие собой интегро-дифференциальные уравнения, для которых аналитически определяются несколько континуальных семейств кусочно-постоянных двухступенчатых решений. Исследуется устойчивость этих решений. Показано существование кусочно-постоянных решений, имеющих более одной точки разрыва. Выполнен численный эксперимент, иллюстрирующий аналитические построения.

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.

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The authors declare that there are no conflicts of interest present.