A cubic system with thirteen limit cycles (Q1018377)

The paper presents a new step in the study of the upper bound for the number of limit cycles \(H(3)\) for planar cubic systems. By using the investigation of zeros of corresponding abelian integrals, the authors prove the following main result: There exists a cubic system having the form \[ \dot{x}=-\frac{\partial H(x,y)}{\partial y} + \varepsilon P(x,y), \qquad \dot{y}=\frac{\partial H(x,y)}{\partial x} + \varepsilon Q(x,y), \] where \(H(x,y)\) is a polynomial of degree \(4\), \(P\) and \(Q\) are polynomials of degree \(3\), which has at least \(13\) limit cycles in the plane for sufficiently small parameter \(\varepsilon\). A previous result of \textit{P. Yu} and \textit{M. Han} [Commun. Pure Appl. Anal. 3, No.~3, 515--525 (2004; Zbl 1085.34028)] demonstrated that \(H(3)\geq12.\)