Infinitely Remote Singularities of Special Differential Dynamic Systems

İrina Andreeva

Research Article

Infinitely Remote Singularities of Special Differential Dynamic Systems

Year 2018, Issue: 4, 1 - 7, 04.12.2018

İrina Andreeva

Abstract

The work is devoted to the results of a fundamental study on the
arithmetical plane of a broad special family of differential dynamic systems
having polynomial right parts. Let those polynomials be a cubic and a square
reciprocal forms. A task of a whole investigation was to find out all
topologically different phase portraits in a Poincare circle and indicate close
to coefficient criteria of them. To achieve this goal a Poincare method of the
central and the orthogonal consecutive displays (or mappings) has been used. As
a rezult more than 250 topologically different phase portraits
in a total have been constructed. Every portrait we depict with a special table
called a descriptive phase portrait. Each line of such a special table
corresponds to one invariant cell of the phase portrait and describes its
boundary, a source of its phase flow and a sink of it. All finite and
infinitely remote singularities of dynamic systems under consideration were
fully investigated. Namely infinitely remote singularities are discussed in the
present article.

Keywords

Dynamic systems , Phase portraits , Phase flows , Poincare sphere , Poincare circle , Singular points , Separatrices , Trajectories

References

Andronov, A.A., Leontovich, E.A., Gordon, I.I., & Maier, A.G. (1973). Qualitative theory of second-order dynamic systems. New York, NY: Wiley. Andreev, A.F., & Andreeva, I.A. (1997). On limit and separatrix cycles of a certain quasiquadratic system. Differential Equations, 33 (5), 702 – 703. Andreev, A.F., & Andreeva, I.A. (2007). Local study of a family of planar cubic systems. Vestnik St. Petersburg University: Ser.1. Mathematics, Mechanics, Astronomy, 2, 11- 16. DOI: 10.3103/S1063454107020021, EID: 2-s2.0-84859730890. Andreev, A.F., Andreeva, I.A., Detchenya, L.V., Makovetskaya, T.V., & Sadovskii, A.P. (2017). Nilpotent Centers of Cubic Systems. Differential Equations, 53(8), 1003 - 1008. DOI: 10.1134/S0012266117080018, EID: 2-s2.0-85029534241. Andreev, A.F., & Andreeva, I.A. (2007). Phase flows of one family of cubic systems in a Poincare circle. I. Differential Equations and Control, 4, 17-26. Andreev, A.F., & Andreeva, I.A. (2008). Phase flows of one family of cubic systems in a Poincare circle. II. Differential Equations and Control, 1, 1 - 13. Andreev, A.F., & Andreeva, I.A. (2008). Phase flows of one family of cubic systems in a Poincare circle. III. Differential Equations and Contro/, 3, 39 - 54. Andreev, A.F., & Andreeva, I.A. (2009). Phase flows of one family of cubic systems in a Poincare circle. Differential Equations and Control, 4, 181 - 213. Andreev, A.F., &Andreeva, I.A. (2010). Phase flows of one family of cubic systems in a Poincare circle. Differential Equations and Control, 4, 6- 17. Andreev, A.F., & Andreeva, I.A. (2017). Investigation of a Family of Cubic Dynamic Systems. Vibroengineering Procedia, 15, 88 – 93. DOI: 10.21595/vp.2017.19389.

Year 2018, Issue: 4, 1 - 7, 04.12.2018

İrina Andreeva

Abstract

References

Andronov, A.A., Leontovich, E.A., Gordon, I.I., & Maier, A.G. (1973). Qualitative theory of second-order dynamic systems. New York, NY: Wiley. Andreev, A.F., & Andreeva, I.A. (1997). On limit and separatrix cycles of a certain quasiquadratic system. Differential Equations, 33 (5), 702 – 703. Andreev, A.F., & Andreeva, I.A. (2007). Local study of a family of planar cubic systems. Vestnik St. Petersburg University: Ser.1. Mathematics, Mechanics, Astronomy, 2, 11- 16. DOI: 10.3103/S1063454107020021, EID: 2-s2.0-84859730890. Andreev, A.F., Andreeva, I.A., Detchenya, L.V., Makovetskaya, T.V., & Sadovskii, A.P. (2017). Nilpotent Centers of Cubic Systems. Differential Equations, 53(8), 1003 - 1008. DOI: 10.1134/S0012266117080018, EID: 2-s2.0-85029534241. Andreev, A.F., & Andreeva, I.A. (2007). Phase flows of one family of cubic systems in a Poincare circle. I. Differential Equations and Control, 4, 17-26. Andreev, A.F., & Andreeva, I.A. (2008). Phase flows of one family of cubic systems in a Poincare circle. II. Differential Equations and Control, 1, 1 - 13. Andreev, A.F., & Andreeva, I.A. (2008). Phase flows of one family of cubic systems in a Poincare circle. III. Differential Equations and Contro/, 3, 39 - 54. Andreev, A.F., & Andreeva, I.A. (2009). Phase flows of one family of cubic systems in a Poincare circle. Differential Equations and Control, 4, 181 - 213. Andreev, A.F., &Andreeva, I.A. (2010). Phase flows of one family of cubic systems in a Poincare circle. Differential Equations and Control, 4, 6- 17. Andreev, A.F., & Andreeva, I.A. (2017). Investigation of a Family of Cubic Dynamic Systems. Vibroengineering Procedia, 15, 88 – 93. DOI: 10.21595/vp.2017.19389.

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Details

Primary Language	English
Subjects	Engineering
Journal Section	Articles
Authors	İrina Andreeva
Publication Date	December 4, 2018
Published in Issue	Year 2018 Issue: 4

Cite

APA	Andreeva, İ. (2018). Infinitely Remote Singularities of Special Differential Dynamic Systems. The Eurasia Proceedings of Science Technology Engineering and Mathematics(4), 1-7.