#### Rebiha Benterki, A class of planar polynomial vector field with explicit non-algebraic limit cycles, Vol. 2020 (2020), Article ID 17, pp. 1-8

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DOI: 10.23952/cot.2020.17

Received March 3, 2020; Accepted July 6, 2020; Published August 1, 2020

Abstract. With the help of the Bernoulli equation, we establish a new class of planar polynomial vector field of the form:
$x^{\prime }=-y(x^{2}+y^{2})^{l}+xR_{2l}(x,y)+xS_{m}(x,\text{ }y),$
$y^{\prime }=x(x^{2}+y^{2})^{l}+yR_{2l}(x,\text{ }y)+yS_{m}(x,\text{ }y),$
where $R_{2l},$ $S_{m}$ are homogeneous polynomials of degrees $2l$ and $m$, respectively with $l. We prove that this class of differential systems has at most one explicit limit cycle. We obtain our result by giving a general class of differential systems of degree five with explicit non algebraic limit cycles.