The maximum number of small-amplitude limit cycles in Liénard-type systems with cubic restoring terms (Q6562450)

The paper is devoted to an investigation of small-amplitude limit cycles for two classes of Liénard systems in the form \[\frac{dx}{dt} = y - F(x,\mu), \quad \frac{dy}{dt} = x-x^2-\frac{1}{k}x^3\] with a non-zero constant \(k\), where the damping term \(F(x,\mu)\) is either a polynomial or a rational function \(\frac{q_n(x)}{p_m(x)}\) with polynomials \(q_n(x)\) and \(p_m(x)\) of degrees \(n\) and \(m\) respectively. The vector parameter \(\mu\) represents the coefficients of the polynomials from \(F(x,\mu)\).\N\NBy utilizing the Picard-Fuchs equation, the author gains the upper bounds for the number of small-amplitude limit cycles in the mentioned two systems for any \(k \neq 0\). Moreover, the smaller upper bounds are derived for \(k<-4\), \(k \neq -9/2\), and the upper bound is sharp if the damping term is polynomial.