Piksasov M. The Geometrical design of elliptic areas of phases portraits of the oscillating systems at determination of their areas of stability.

The Object - a geometric manifestations to stability of the fluctuations of the mechanical systems with using their phase portrait; the purpose - a development graphoanalitic way of the discovery on phase portrait of the oscillatory system to families spiral crooked, each of which is aproximated by ellipse with the further choice of the ellipse maximum area, which is used for automatic management oscillatory system on criterion not output it from area of stability; the methods - a theory of the differential equations and computing mathematicians, elements to theories of the fluctuations, dynamic systems and computing mathematicians, as well as elements computer animation graphs in ambience of the mathematical processor Maple; theoretical and practical results - is designed way to aproximations ellipse spiral of the phase crooked oscillatory systems by means of moving in space parameter with determination of the ellipse maximum area; the designed way of the description of the ellipse of the general provisions with the centre at the beginning initially coordinates on coordinate its point osculations with coordinate frame; the advanced way to elliptical localization of the firm focus of the differential equations; the presented variants to hardware realization managerial system with sensor of the change the corner of the driftage and velocities рыс-кания for hovercraft; scientific novelty - in work is received new decision scientifically-practical problem to localizations by ellipse on phase portrait of the differential equation of the oscillatory system rack focus by by discovery limiting ellipse, chosen amongst ensemble ellipse, which aproximate reconverging in this focus spiral phase curves, which are revealled by means of computer animation depending on change the dominant parameter; the degree of the