The Maximum Number of Small-Amplitude Limit Cycles in Liénard-Type Systems with Cubic Restoring Terms

In this paper, we investigate the small-amplitude limit cycles of two classes of Liénard systems of the form \(\dot{x}=y-F(x, \mu ), \dot{y}=-g(x)\) $\dot{x}=y-F(x,\mu ),\dot{y}=-g(x)$ , where the damping term \(F(x, \mu )\) $F(x,\mu )$ is either a polynomial or a rational function \(\frac{q_{n}(x)}{p_{m}(x)}\) $\frac{q_n(x)}{p_m(x)}$ , ( \(q_{n}(x)\) $q_n(x)$ and \(p_{m}(x)\) $p_m(x)$ are polynomials in x with degrees n and m, respectively), \(\mu \) $\mu$ represents coefficients and the cubic restoring term \(g(x)=x-x^{2}-\frac{1}{k}x^{3}\) $g(x)=x-x^2-\frac{1}{k}{x}^3$ with a non-zero constant k. By utilizing Picard-Fuchs equation, we gain the upper bounds of the number of small-amplitude limit cycles of these two systems for any \(k\ne 0\) $k\ne 0$ . Moreover, the smaller upper bounds are obtained for \(k<-4\) $k<-4$ , \(k\ne -\frac{9}{2}\) $k\ne -\frac{9}{2}$ , and the upper bound is sharp if the damping term is polynomial.