Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Phase Flows of One Family of Cubic Systems in a Poincare Circle. I.

Author(s):

A. F. Andreev

Russia, 198504, Petergof, Universitetski pr.28
S.Peterburg State University
Faculty of Mathematics and Mechanics

irandr@inbox.ru

I. A. Andreeva

Russia, 195251, Saint-Petersburg, Politekhnicheskaya str., 29,
Saint-Petersburg state technic university,
dept. of Higher mathematics

irandr@inbox.ru

Abstract:

On a real (x,y)-plane a normal system of ordinary differential equations is considered, being the right parts are respectively quadratic and cubic forms of x and y with arbitrary constant coefficients. The task is to investigate the behavior of its trajectories on an extended (x,y)- plane or, in equivalent terms, in a Poincare circle. In Part I of this investigation all possible for this system topological types of a singular point O(0,0) are revealed, and criteria of their realization are formulated. Topological types of the point O are given:

• in terms of bundles of O-semiorbits of the system of N (node) and S (saddle) types, and
• in terms of O-sectors of Bendixon of E (elliptic), H (hyperbolic) and P (parabolic) types.

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