About isochrons of strong focus

Authors

  • Alexander E. Rudenok Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus
  • Mikhail N. Vasilevich Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus

Keywords:

centre, weak focus, strong focus, isochronism, isochrones, Poincaré – Dulac normal form, system with quadratic nonlinearities

Abstract

The article analyses the methods for studying the isochronism of autonomous systems on a plane for different types of singular points: centre, weak focus and strong focus. The difficulty of determining a weak isochronous focus is explained. Examples of systems in the Poincaré – Dulac normal form, which show that the requirement to be isochronous imposes weak restrictions on a system with a weak focus, are given. A theorem on the form of a normalising change of a strongly isochronous strong focus is proved. Isochrones of a strongly isochronous strong focus of a system with quadratic nonlinearities are found. Phase portraits of the obtained isochronous foci and their isochrones are constructed.

Author Biographies

  • Alexander E. Rudenok, Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus

    PhD (physics and mathematics), docent; associate professor at the department of differential equations and system analysis, faculty of mechanics and mathematics

  • Mikhail N. Vasilevich, Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus

    PhD (physics and mathematics), docent; associate professor at the department of differential equations and system analysis, faculty of mechanics and mathematics

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Published

2025-10-13

Issue

No. 2 (2025): Journal of the Belarusian State University. Mathematics and Informatics

Section

Differential Equations and Optimal Control

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How to Cite

[1]
Rudenok, A.E. and Vasilevich, M.N. 2025. About isochrons of strong focus. Journal of the Belarusian State University. Mathematics and Informatics. 2 (Oct. 2025), 16–29. DOI:https://doi.org/10.33581/2520-6508-2025-2-16-29.