The existence of a three-dimensional center in a system of three differential equations

About the authors:

Fedor S. Bayanov, Master Student, University of Tyumen; fedyabay@yandex.ru

Tatyana E. Kazantseva, Senior Lecturer, Department of Fundamental Mathematics and Mechanics, University of Tyumen; eLibrary AuthorID, t.e.kazanceva@utmn.ru

Vladislav V. Machulis, Cand. Sci. (Ped.), Associate Professor, Department of Fundamental Mathematics and Mechanics, University of Tyumen; eLibrary AuthorID, ORCID, ResearcherID, mareliks@gmail.com

Abstract:

This article studies a system of three differential equations with cubic polynomials in the right-hand sides and seven arbitrary parameters. The authors aim to meet the conditions for the parameters which will allow the system to have a center at the origin. An area of the origin is completely foliated by periodic orbits at such system. This corresponds to the low-amplitude periodic solutions. By introducing a small parameter and transformation to new coordinates, the system is reduced to a set of two differential equations. The solution of the transformed system is represented as a series in powers of a small parameter, as is the Poincare map and displacement map. The study of these maps obtains conditions for the existence of a three-dimensional center.

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