Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)5192410.22363/2658-4670-2026-34-2-226-240JFRCXFModeling and SimulationМатематическое моделированиеResearch ArticleOn the behavior of orbits of Vanhaecke system on integral surfacesО поведении орбит системы Ванхекке на интегральных поверхностяхhttps://orcid.org/0009-0007-6504-8370WangShiweiВанШ.

PhD Student of Department of Computational Mathematics and Artificial Intelligence, RUDN University

1995wsw@gmail.comhttps://orcid.org/0000-0001-6541-66036602318510P-8123-2016MalykhMikhail D.МалыхМ. Д.

DSc., Head of Department of Computational Mathematics and Artificial Intelligence, RUDN University; Senior Researcher of Joint Institute for Nuclear Research

malykh-md@rudn.ruhttps://orcid.org/0000-0002-1856-46438783969400B-8497-2016SevastianovLeonid A.СевастьяновЛ. А.

DSc., Professor of Department of Computational Mathematics and Artificial Intelligence, RUDN University; Senior Researcher of Joint Institute for Nuclear Research

sevastianov-la@rudn.ruhttps://orcid.org/0000-0002-5721-4558ZorinAleksander V.ЗоринА. В.

DSc., Professor of Department of Computational Mathematics and Artificial Intelligence of RUDN University

zorin-av@rudn.ruRUDN UniversityРоссийский университет дружбы народовJoint Institute for Nuclear ResearchОбъединённый институт ядерных исследований15082026342VOL 34, NO2 (2026)ТОМ 34, №2 (2026)22624020082026Copyright ©; 2026, Wang S., Malykh M.D., Sevastianov L.A., Zorin A.V.Copyright ©; 2026, Ван Ш., Малых М.Д., Севастьянов Л.А., Зорин А.В.2026Wang S., Malykh M.D., Sevastianov L.A., Zorin A.V.Ван Ш., Малых М.Д., Севастьянов Л.А., Зорин А.В.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/51924

In the 1990s, P. Vanhecke described a Hamiltonian system with two degrees of freedom and a polynomial Hamiltonian integrable in Abelian functions of two variables. This system provides a convenient example of an integrable system (in the sense of Liouville) in which integral curves are wound on a two-dimensional manifold, an algebraic surface in a 4-dimensional phase space. We show that all necessary calculations can be performed in the Sage system. The results of numerical experiments performed in our package FDM for Sage are presented. It is shown that orbits of Vanhecke system can be divided into two classes, periodic and non-periodic. The points of the periodic orbits corresponding to solutions with the same period form an equiperiodic surface. There are infinitely many different algebraic equiperiodic surfaces. The behavior of nonperiodic orbits in numerical experiments is determined not so much by the rationality or irrationality of the ratio of two periods of Abelian functions, but by the possibility of approximating this number with a rational fraction with high accuracy.

В 1990-х годах П. Ванхекке описал гамильтоновую систему с двумя степенями свободы и полиномиальной гамильтоном, которая интегрируема в абелевы функции двух переменных. Эта система доставляет удобный пример вполне интегрируемой системы, в которой интегральные кривые навиты на двумерное многообразие - алгебраическую поверхность в 4-мерном фазовом пространстве. В статье показано, что все необходимые расчёты могут быть выполнены в системе Sage. Представлены результаты численных экспериментов, выполненных в нашем пакете FDM для Sage. Показано, что орбиты системы Ванхекке можно разделить на два класса: периодические и непериодические. Точки периодических орбит, соответствующие решениям с одинаковым периодом, образуют эвкипериодическую поверхность, причём существует бесконечно много различных алгебраических эквипериодических поверхностей. Поведение непериодических орбит в численных экспериментах определяется не столько рациональностью или иррациональностью соотношения двух периодов абелевых функций, сколько возможностью аппроксимации этого числа рациональной дробью с высокой точностью.

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