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.
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.
Details
| Primary Language | English |
|---|---|
| Subjects | Engineering |
| Journal Section | Articles |
| Authors | |
| Publication Date | December 4, 2018 |
| Published in Issue | Year 2018Issue: 4 |
