More limit cycles than expected in Liénard equations

The paper deals with classical polynomial Liénard equations, i.e. planar vector fields associated to scalar second order differential equations where is a polynomial. We prove that for a well-chosen polynomial of degree the equation exhibits limit cycles. It induces that for there exist polynomials of degree such that the related equations exhibit more than limit cycles. This contradicts the conjecture of Lins, de Melo and Pugh stating that for Liénard equations as above, with of degree the maximum number of limit cycles is 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 Here, is a well-chosen family of polynomials of degree with parameter and is a small positive parameter tending to 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 . 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.