Abstract

A lot of partial results are known about the Liénard differential equations: $x\prime =y-F$_aⁿ(x), y′=-x. Here $F$_aⁿ is a polynomial of degree $2n+1$, $F$_aⁿ(x)=Σ_i=1²ⁿa_ixⁱ+x²ⁿ⁺¹ where $a=(a$₁,...,a_2n)∈R²ⁿ.
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 half-axis $Ox\; =x\ge 0$, and that this map is analytic and repulsive 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 slow-fast 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 related to the finiteness of the number of limit cycles which bifurcate from slow-fast cycles of these singular equations.