The radical of the focal values ideal of the complex Kukles system

  • Anton P. Sadovskii Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus
  • Tatsiana V. Makavetskaya Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus
  • Dmitry N. Cherginets Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

Abstract

In the article it is considered the center-focus problem for complex Kukles system x = y, y = -x + Ax2 + 3Bxy + Cy2 + Kx3 + 3Lx2y + Mxy2 + Ny3. The problem is solved by the new method, obtained by A. P. Sadovski and based on the method of normal forms. Instead of investigating the variety of ideal of focal values it is proposed to study the variety of ideal with the basis – polynomials obtained by a new method. The study of the radical of such ideal is divided into two parts: the trivial case where BN = 0, and the case of BN ≠ 0. If BN = 0 it is obtained five series of center conditions for the complex system, in particular, four series of center conditions for the real system. In the case of BN ≠ 0 it can be assumed B = N in the Kuklesʼ system. This assumption simplifies the further study of this case (it is obtained three series of the existence of the complex center, in particular, two series of the existence of the real center). Thus, as a result of research in the present paper it is presented the necessary and sufficient conditions for the complex and real centers existence for the complex and real Kuklesʼ systems respectively. 

Author Biographies

Anton P. Sadovskii, Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

doctor of science (physics and mathematics), full professor; professor at the department of differential equations and systems analysis, faculty of mechanics and mathematics

Tatsiana V. Makavetskaya, Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

PhD (physics and mathematics); associate professor at the department of analytical economics and econometrics, faculty of economics

Dmitry N. Cherginets, Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

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

References

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Published
2018-01-23
Keywords: center-focus problem, Kukles system, complex center variety, focal values, normal forms, radical of ideal
How to Cite
Sadovskii, A. P., Makavetskaya, T. V., & Cherginets, D. N. (2018). The radical of the focal values ideal of the complex Kukles system. Journal of the Belarusian State University. Mathematics and Informatics, 2, 4-11. Retrieved from https://journals.bsu.by/index.php/mathematics/article/view/742
Issue
No 2 (2017): Journal of the Belarusian State University. Mathematics and Informatics
Section
Differential Equations and Optimal Control

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