Center conditions for generalized polynomial Kukles systems

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2017
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Giné, JaumeGiné, Jaume - ORCID ID
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In this paper we study the center problem for certain generalized Kukles systems \[ \dot{x}= y, \qquad \dot{y}= P_0(x)+ P_1(x)y+P_2(x) y^2+ P_3(x) y^3, \] where $P_i(x)$ are polynomials of degree $n$, $P_0(0)=0$ and $P_0'(0) <0$. Computing the focal values and using modular arithmetics and Gr\'{o}bner bases we find the center conditions for such systems when $P_0$ is of degree $2$ and $P_i$ for $i=1,2,3$ are of degree $3$ without constant terms. We also establish a conjecture about the center conditions for such systems.
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Communications On Pure And Applied Analysis, 2017, vol. 16, núm. 2, p. 417-425