In this paper, we study a second-order linear system with a differentiating operator D and constant coefficients, defined along a helical line on a cylindrical surface. The work is also based on fundamental studies of the almost periodic solutions of ordinary differential equations and partial differential equations as described in the works of V.Kh. Kharasakhal and D.U.Umbetzhanov. The method of integration by periodic characteristics of the diagonal differentiation operator for multiperiodic functions, proposed by Zh.A. Sartabanov, is widely used here. This method allows us to generalize the classical concept of a center to systems in a spirally symmetric space and to study their dynamic properties. Information about the spiral characteristic of the operator D, described by the properties of the arc and cut-off functions, is presented, and the concept of a center is introduced. Additionally, the point of time (τ,t) moves along a helical line on the surface of a vertical circular cylinder, and the plane (х₁, х₂) defines the phase space. If the solutions of the system under consideration (τ, t) are periodic, with radius-vector and angular-phase components, an expression is defined that determines that point (0, 0) will be the focal point. If there is a small linear deviation in the system’s matrix, it is proved that the center transforms into a focus. During the course of the study, the conditions for preserving the center were determined, as well as the parameters that cause it to become a focus.

SARTABANOV ZH.А.

Doctor of Physico-Mathematical Sciences, Professor, K. Zhubanov Aktobe Regional University, Aktobe, Kazakhstan.

Е-mail: sartabanov42@mail.ru, https://orcid.org/0000-0003-2601-2678

OMAROVA B.Z.

PhD, K. Zhubanov Aktobe Regional University, Aktobe, Kazakhstan.

Е-mail: bomarova@zhubanov.edu.kz, https://orcid.org/0000-0002-3267-2501

TULEUOVA M.K.

Master’s student, K. Zhubanov Aktobe Regional University, Aktobe, Kazakhstan.

Е-mail: tuleuova_medina99@mail.ru, https://orcid.org/0009-0005-4434-239X 

1. Харасахал В.Х. Почти-периодические решения обыкновенных дифференциальных уравнений / В.Х. Харасахал. - Алма-Ата: Наука, 1970. – 200 с.
2. Умбетжанов Д.У. Почти многопериодические решения дифференциальных уравнений в частных производных / Д.У. Умбетжанов. - Алма-Ата: Наука, 1979. – 210 с.
3. Сартабанов Ж.А. Об одном спососбе изучения периодических решений уравнений в частных производных специального вида / Ж.А. Сартабанов // Изв. АН КазССР. Серия физ.-мат. - 1989. - №1. - C. 42-48.
4. Сартабанов Ж.А. Периодты функциялар және кейбір қарапайым дифференциалдық теңдеулердің периодты шешімдері / Ж.А. Сартабанов. – Алматы: РБК, 2001. – 108 б.
5. Сартабанов Ж.А. Периодичность характеристик оператора дифференцирования по диагонали / Ж.А. Сартабанов // Вестник КазНПУ им.Абая, серия «Физико-математические науки». - 2023. – Т. 82., №2. – C. 40-53.
6. Sartabanov Zh. Integrating multiperiodic functions along the periodic characteristics of the diagonal differentiation operator / Zh. Sartabanov, B. Omarova, G. Aitenova, A. Zhumagaziyev // KazNU Bulletin. Mathematics, Mechanics, Computer Science Series. - 2023. - Vol. 120, №4. - P. 52-68. DOI: https://doi.org/10.26577/JMMCS2023v120i4a6

cylindrical surface, helix line, differentiation operator, multiperiodicity, focus, center