More limit cycles than expected in Lienard equations

Abstract

The paper deals with classical polynomial Lienard 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 >= 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 Lienard 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 Lienard equations. More precisely we find our example inside a family of second order differential equations ex" + f mu(x)x' + x = 0. Here, f mu is a well-chosen family of polynomials of degree 6 with parameter mu is an element of R-4 and e 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 epsilon = 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.