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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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##### References:
 [1] Chen, L.; Wang, M., The relative position and the number of limit cycles of a quadratic differential system, Acta math. sinica, 22, 751-758, (1979), (in Chinese) · Zbl 0433.34022 [2] Christopher, C.; Li, C., Limit cycles of differential equations, (2007), Birkhäuser Verlag [3] Christopher, C.; Lloyd, N.G., Polynomial systems: A lower bound for the Hilbert numbers, Proc. R. soc. lond. ser. A, 450, 1938, 218-224, (1995) · Zbl 0839.34033 [4] Dumortier, F.; Li, C., Perturbation from an elliptic Hamiltonian of degree four. III. global centre, J. differential equations, 188, 473-511, (2003) · Zbl 1056.34044 [5] Dumortier, F.; Li, C., Perturbation from an elliptic Hamiltonian of degree four. IV. figure eight-loop, J. differential equations, 188, 512-554, (2003) · Zbl 1057.34015 [6] Li, J., Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Int. J. bifur. chaos, 13, 47-106, (2003) · Zbl 1063.34026 [7] Li, J.; Huang, Q., Bifurcations of limit cycles forming compound eyes in the cubic system, Chinese ann. math. ser. B, 8, 391-403, (1989) · Zbl 0658.34020 [8] Shi, S., A concrete example of the existence of four limit cycles for quadratic system, Sci. sinica, 23, 153-158, (1980) · Zbl 0431.34024 [9] Yu, P.; Han, M., Twelve limit cycles in a cubic case of the 16th Hilbert problem, Int. J. bifur. chaos, 15, 2191-2205, (2005) · Zbl 1092.34524
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