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Putting a boundary to the space of Liénard equations

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  • 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 half-axis $Ox=\{x \geq 0\},$ 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 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 be related to the finiteness of the number of limit cycles which bifurcate from slow-fast cycles of these singular equations.
    Mathematics Subject Classification: Primary: 34C05; Secondary: 34C26.


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