Conference publications


XVI conference

Reprojection phase trajectories. Phase portraits in Poincare's circle

Vdovina E.V.

The Ural state university the A.M.Gorkji, Matematik-mechanical faculty, faculty of calculus mathematics, Russia, 620078, Ekaterinburg, street Pedagogical 13 - 24. Ph.: (343)374-63-79, E-mail:

1 pp. (accepted)

As a rule, in the university tutorials devoted to the ordinary differential equations, classification of versions of phase portraits for linear homogeneous systems with constant factors II of the order is spent. But, in our opinion, in one of them the statement of this subject is not possible before full end. Namely: the phase portrait in a final part of a plane is under construction and on it the point is put. And the behaviour of phase trajectories on infinity is not examined, and as the family of integrated curves of system is not restored by the form phase trajectories[1].

For linear homogeneous system II of the order with constant factors it is possible to lead. Poincare's transformations which allow to display a phase plane of system in a circle of individual radius with the center in the beginning of coordinates - Poincare's circle, and to indefinitely removed points of a plane there correspond{meet} the points laying on border of a circle [2].

Classification of positions of balance of the system, arising on border of a circle is lead. With the help of software package ОDЕ phase portraits of system, both on a plane, and in Poincare's circle are under construction.

With use machine schedules from a phase portrait to family of integrated curves can be made transition rather evident. If the phase portrait for independent system II of the order can be received by designing family of integrated curves along axis Оt on a phase plane хОу for transition from a phase portrait to family of integrated curves it is possible to use return process - process reprojection. In three-dimensional space the cylinder in which for directing the phase trajectory undertakes is under construction, and as forming the straight lines parallel to an axis 0t are examined. The integrated curves laying on the cylinder, are under construction on a phase trajectory in view of character of stability of position of balance to which she adjoins.

For an illustration of this process special programs have been made.

The literature.

1. Pontrjagin A.S., The Ordinary differential equations, М., "Science", 1970,331pag.

2. Andronov A.A., Leontovich E.A., Gordon I.I., Majer A.G., the Qualitative theory of dynamic systems of the second order, М., "Science", 1966, 562 pag.

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