2007 | OriginalPaper | Buchkapitel
Hilbert’s 16th Problem and Its Weak Form
Erschienen in: Limit Cycles of Differential Equations
Verlag: Birkhäuser Basel
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Consider the planar differential systems
(1.1)
$$ \dot x = P_n (x,y),\dot y = Q_n (x,y), $$
where
P
n
and
Q
n
are real polynomials in
x, y
and the maximum degree of
P
and
Q
is
n
. The second half of the famous Hilbert’s 16th problem, proposed in 1900, can be stated as follows (see [70]):
For a given integer n, what is the maximum number of limit cycles of system
(1.1)
for all possible P
n
and Q
n
? And how about the possible relative positions of the limit cycles ?