Investigation Methods for a Family of Cubic Dynamic Systems

Abstract

A broad family of differential dynamic systems is considered ona real plane of their phase variables x, y. The main common feature of systemsunder consideration is the follows: every particular system includes twoequations with polynomial right parts of the third order in one equation and ofthe second order in another one. These polynomials are mutually reciprocal inthe following understanding: their decomposition into forms of lower order doesnot contain common multipliers. The whole family of such dynamic systems hasbeen split into subfamilies according to numbers of different multipliers inthe abovementioned decomposition and depending on an order of sequence ofdifferent roots of polynomials. Every subfamily has been studied in a Poincarecircle using especially developed investigation methods. As a result allpossible for the dynamic systems belonging to this family phase portraits havebeen revealed and described. There appeared to exist more than 200 differenttopological types of phase portraits in a Poincare circle. The obtained resultshave a scientific interest as well as a methodical and educational one.