At the eighth of August 1900 impressive German mathematician David Hilbert brought a conversation "Mathematical difficulties" on the moment Interna tional Congress of Mathematicians in Paris. the controversy coated essentially all instructions of mathematical considered that point and contained an inventory of 23 difficulties which decided the additional improvement of mathema tics in lots of respects (1, 119]. Hilbert's 16th challenge (the moment half) used to be acknowledged as follows: challenge. to discover the utmost quantity and to figure out the relative place of restrict cycles of the equation dy Qn(X, y) -= dx Pn(x, y)' the place Pn and Qn are polynomials of actual variables x, y with genuine coeffi cients and never more than n measure. The examine of restrict cycles is a fascinating and extremely tricky challenge of the qualitative conception of differential equations. This concept was once origi nated on the finish of the 19th century within the works of 2 geniuses of the realm technology: of the Russian mathematician A. M. Lyapunov and of the French mathematician Henri Poincare. A. M. Lyapunov set forth and solved thoroughly within the very broad category of instances a different challenge of the qualitative thought: the matter of movement balance (154]. In flip, H. Poincare acknowledged a normal challenge of the qualitative research which used to be formulated as follows: now not integrating the differential equation and utilizing merely the homes of its right-hand aspects, to offer as extra as possi ble whole details at the qualitative behaviour of indispensable curves outlined by means of this equation (176].
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