On the 8th of August 1900 outstanding German mathematician David
Hilbert delivered a talk "Mathematical problems" at the Second
Interna tional Congress of Mathematicians in Paris. The talk
covered practically all directions of mathematical thought of that
time and contained a list of 23 problems which determined the
further development of mathema tics in many respects (1, 119].
Hilbert's Sixteenth Problem (the second part) was stated as
follows: Problem. To find the maximum number and to determine the
relative position of limit cycles of the equation dy Qn(X, y) -= dx
Pn(x, y)' where Pn and Qn are polynomials of real variables x, y
with real coeffi cients and not greater than n degree. The study of
limit cycles is an interesting and very difficult problem of the
qualitative theory of differential equations. This theory was origi
nated at the end of the nineteenth century in the works of two
geniuses of the world science: of the Russian mathematician A. M.
Lyapunov and of the French mathematician Henri Poincare. A. M.
Lyapunov set forth and solved completely in the very wide class of
cases a special problem of the qualitative theory: the problem of
motion stability (154]. In turn, H. Poincare stated a general
problem of the qualitative analysis which was formulated as
follows: not integrating the differential equation and using only
the properties of its right-hand sides, to give as more as possi
ble complete information on the qualitative behaviour of integral
curves defined by this equation (176]."
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