More limit cycles than expected in Liénard equations

Abstract:

The paper deals with classical polynomial Liénard equations, i.e. planar vector fields associated to scalar second order differential equations $x''+f(x)x’+ x=0$ where $f$ is a polynomial. We prove that for a well-chosen polynomial $f$ of degree $6,$ the equation exhibits $4$ limit cycles. It induces that for $n\geq 3$ there exist polynomials $f$ of degree $2n$ such that the related equations exhibit more than $n$ limit cycles. This contradicts the conjecture of Lins, de Melo and Pugh stating that for Liénard equations as above, with $f$ of degree $2n,$ the maximum number of limit cycles is $n.$ The limit cycles that we found are relaxation oscillations which appear in slow-fast systems at the boundary of classical polynomial Liénard equations. More precisely we find our example inside a family of second order differential equations $\varepsilon x''+f_\mu (x)x’+x=0.$ Here, $f_\mu$ is a well-chosen family of polynomials of degree $6$ with parameter $\mu \in \mathbb {R}^4$ and $\varepsilon$ is a small positive parameter tending to $0.$ We use bifurcations from canard cycles which occur when two extrema of the critical curve of the layer equation are crossing (the layer equation corresponds to $\varepsilon =0)$. As was proved by Dumortier and Roussarie (2005) these bifurcations are controlled by a rational integral computed along the critical curve of the layer equation, called the slow divergence integral. Our result is deduced from the study of this integral.