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A cubic system with thirteen limit cycles. (English) Zbl 1176.34037
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 P. Yu and M. Han [Commun. Pure Appl. Anal. 3, No. 3, 515–525 (2004; Zbl 1085.34028)] demonstrated that \(H(3)\geq12.\)

MSC:
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
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