Investigation Methods for a Family of Cubic Dynamic Systems

Abstract

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.

Keywords

• Dynamic systems,Differential equations,Poincare circle,Phase portraits

References

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