
Abstract
A lot of partial results are known about the Liénard
differential equations : $\dot x= y F_a^n(x),\ \ \dot y =x.$
Here $F_a^n$ is a polynomial of degree $2n+1,\ \ F_a^n(x)=
\sum_{i=1}^{2n}a_ix^i+x^{2n+1},$ where $a = (a_1,\cdots,a_{2n})
\in \R^{2n}.$ For instance, it is easy to see that for any $a$
the related vector field $X_a$ has just a finite number of limit
cycles. This comes from the fact that $X_a$ has a global return
map on the halfaxis $Ox=\{x \geq 0\},$ and that this map is
analytic and repelling at infinity. It is also easy to verify that
at most $n$ limit cycles can bifurcate from the origin. For these
reasons, Lins Neto, de Melo and Pugh have conjectured that the
total number of limit cycles is also bounded by $n,$ in the whole
plane and for any value $a.$
In fact it is not even known if there
exists a finite bound $L(n)$ independent of $a,$ for the
number of limit cycles. In this paper, I want to investigate this
question of finiteness. I show that there exists a finite bound
$L(K,n)$ if one restricts the parameter in a compact $K$ and that
there is a natural way to put a boundary to the space of Liénard
equations. This boundary is made of slowfast equations of
Liénard type, obtained as singular limits of the Liénard
equations for large values of the parameter. Then the existence
of a global bound $L(n)$ can be related to the finiteness of the
number of limit cycles which bifurcate from slowfast cycles
of these singular equations.
Mathematics Subject Classification: Primary: 34C05; Secondary: 34C26.
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