1683-3414 1814-0807 Владикавказский математический журнал Vladikavkaz Mathematical Journal Южный математический институт - филиал Федерального государственного бюджетного учреждения науки Федерального научного центра «Владикавказский научный центр Российской академии наук» (ЮМИ ВНЦ РАН) Analysis of the Existence of Periodic Solutions of the Systems of Nonlinear Differential Equations with a Small Parameter Исследование существования периодических решений систем нелинейных дифференциальных уравнений с малым параметром 10.46698/r6381-0860-2384-e 17176 09 2025 27 3 28 39 https://vmj.ru/eng/archive/detail.php?ELEMENT_ID=17225&SECTION_ID=636 Гришанина Г. Э. Grishanina G. E. anora66@mail.ru Мухамадиев Э. М. Muhamadiev E. M. emuhamadiev@rambler.ru Нуров И. Дж. Nurov I. J. nid1@mail.ru Шарифзода З. И. Sharifzoda Z. I. sakhara-2803@mail.ru Государственный университет Дубна, Россия, 141980, Дубна, ул. Университетская, 19 State University Dubna, 19 Universitetskaya St., Dubna 141980, Russia Вологодский государственный университет, Россия, 160000, Вологда, ул. Ленина, 15 Vologda State University, 15 Lenina St., Vologda 160000, Russia Таджикский национальный университет, Таджикистан, 734063, Душанбе, пр. Рудаки, 17 Tajik National University, 17 Rudaki Ave., Dushanbe 734063, Tajikistan We consider the existence of periodic solutions-cycles in nonlinear differential equations with a small parameter. We obtain necessary and sufficient conditions for the existence of periodic solutions. These conditions significantly expand the applicability of the Pontryagin small parameter method from the theory of dynamical systems on the plane. We do not assume the differentiability of all functions involved in the system. Moreover, the system is not Hamiltonian. In order to prove the existence of periodic solutions of the system of nonlinear differential equations we use topological methods of nonlinear analysis. Based on the proposed methods, we formulate and prove theorems on the necessary and sufficient conditions for the existence of periodic solutions under the condition of continuity of all functions involved in the system. Moreover, we use the transition to the polar coordinate system and Jordan transformations. In the last section we propose a method for developing examples for a specific class of functions. Furthermore, we give an example of a system such that we easy verify the conditions for the existence of periodic solutions for small values of \(\varepsilon\). В работе изучается вопрос о существовании периодических решений-циклов в нелинейных дифференциальных уравнениях с малым параметром. Получены необходимые и достаточные условия существования периодических решений, которые существенно расширяют область применимости метода малого параметра Л. С. Понтрягина из теории динамических систем на плоскости. При этом не предполагается дифференцируемость всех входящих в систему функций, а также того, что система является гамильтоновой. Для доказательства существования периодических решений системы нелинейных дифференциальных уравнений в работе применяются топологические методы нелинейного анализа. На основе предложенных методов сформулированы и установлены теоремы о необходимых и достаточных условиях существования периодических решений при условии непрерывности всех входящих в систему функций. С целью упрощения изучаемой системы в работе используется переход к полярной системе координат и жордановы преобразования. В заключительной части предложен метод разработки примеров для конкретного класса функций, а также приведен пример системы, для которой легко проверяются условия существования периодических решений при малых значениях \(\varepsilon\). нелинейные дифференциальные уравнения малый параметр жорданово преобразование гомотопия вращение векторных полей nonlinear differential equations small parameter Jordan transformation homotopy rotation of vector fields 1 Poincare, H. Selected Works, Moscow, Nauka, 1971, 771 p. (in Russian). Poincare, H. Selected Works, Moscow, Nauka, 1971, 771 p. (in Russian). 2 Lyapunov, A. M. Obshchie zadacha ob ustojchivosti dvizheniya [General Problem of the Stability of Motion], Moscow, State Publishing House of Technical and Theoretical Literature, 1950, 472 p. (in Russian). Lyapunov, A. M. Obshchie zadacha ob ustojchivosti dvizheniya [General Problem of the Stability of Motion], Moscow, State Publishing House of Technical and Theoretical Literature, 1950, 472 p. 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