vestnikmephiВестник НИЯУ МИФИVestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI"2304-487XNational Research Nuclear University "MEPhI"10.1134/S2304487X20040100vestnikmephi-99Research ArticleДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ДИНАМИЧЕСКИЕ СИСТЕМЫDIFFERENTIAL EQUATIONS AND DYNAMIC SYSTEMSАналитические свойства решений трехмерных автономных консервативных систем с одной или тремя квадратичными нелинейностями без хаотического поведенияAnalytical Properties of Solutions of Three-Dimensional Autonomous Conservative Systems with One or Three Quadratic Nonlinearities without Chaotic BehavioЦегельникВ. В.Tsegel’nikV. V.

220013

Минск

220013

Minsk

tsegvv@bsuir.byБелорусский государственный университет информатики и радиоэлектроникиБеларусьBelarusian State University of Informatics and RadioelectronicsBelarus20201602202394338344Copyright © Цегельник В.В., 20232023Цегельник В.В.Tsegel’nik V.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnikmephi.elpub.ru/jour/article/view/99

   Analytical properties of solutions of four families of three-dimensional autonomous conservative systems with one or three quadratic nonlinearities are investigated. Conservative systems of the first family have one quadratic nonlinearity and three linear components. Systems of the second family are conservative systems that have one quadratic nonlinearity, two linear components, and one constant. Conservative systems of the third family contain three quadratic nonlinearities and one linear component. Systems of the fourth family include conservative systems with three quadratic nonlinearities and one constant term. The absence of chaos in these systems is their common qualitative property. To analyze the solutions of the considered systems, Painlevé test, as well as the reduction of the systems to their equivalent equations of the second or third order and the comparison of the latter with the known nonlinear P-type equations, has been used. The systems whose general solutions have the Painlevé properties are highlighted. The solutions of such systems are expressed in terms of elliptic functions or solutions of the first Painlevé equation. It is shown that the systems whose general solutions contain moving critical points include those in which one of the component has no moving singular points at all. The systems considered in this work belong to a class of seven families of conservative systems (with the total number of members in the right parts equal to 4). The analytical properties of solutions of the systems of the other three families will be studied elsewhere.

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