Investigation Methods for a Family of Cubic Dynamic Systems

Yıl 2018, Sayı: 2, 439 - 446, 19.08.2018

İrina Andreeva

Öz

A broad family of differential dynamic systems is considered on
a real plane of their phase variables x, y. The main common feature of systems
under consideration is the follows: every particular system includes two
equations with polynomial right parts of the third order in one equation and of
the second order in another one. These polynomials are mutually reciprocal in
the following understanding: their decomposition into forms of lower order does
not contain common multipliers. The whole family of such dynamic systems has
been split into subfamilies according to numbers of different multipliers in
the abovementioned decomposition and depending on an order of sequence of
different roots of polynomials. Every subfamily has been studied in a Poincare
circle using especially developed investigation methods. As a result all
possible for the dynamic systems belonging to this family phase portraits have
been revealed and described. There appeared to exist more than 200 different
topological types of phase portraits in a Poincare circle. The obtained results
have a scientific interest as well as a methodical and educational one.

Anahtar Kelimeler

Dynamic systems, Differential equations, Poincare circle, Phase portraits

Kaynakça

• 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.