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Using the averaging theory of first and second order we study the
maximum number of limit cycles of the polynomial differential
systems
$
\dot x=y, \qquad \dot y= -x-\epsilon(h_{1}(x)+p_{1}(x)y+q_{1}(x)y^{2})-\epsilon
^{2}(h_{2}(x)+p_{2}(x)y+q_{2}(x)y^{2}),
$
which bifurcate from the periodic orbits of the linear center $\dot
x= y$, $\dot y= -x$, where $\epsilon$ is a small parameter. If the degrees
of the polynomials $h_1$, $h_2$, $p_1$, $p_2$, $q_1$ and $q_2$ are equal to $n$,
then we prove that this maximum number is $[n/2]$ using the
averaging theory of first order, where $[\cdot]$ denotes the integer
part function; and this maximum number is at most $n$ using the
averaging theory of second order. |
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