# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021299
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## The cyclicity of a class of quadratic reversible centers defining elliptic curves

 School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, China

*Corresponding author: Changjian Liu

Received  August 2021 Revised  November 2021 Early access December 2021

Fund Project: The work was supported by the NSF of China (No.11771315)

In this paper, the cyclicity of period annulus of an one-parameter family quadratic reversible system under quadratic perturbations is studied which is equivalent to the number of zeros of any nontrivial linear combination of three Abelian integrals. By the criteria established in [28] and the asymptotic expansions of Abelian integrals, we obtain that the cyclicity is two when the parameter in $(-\infty,-2)\cup[-\frac{8}{5},+\infty)$. Moreover, we develop new criteria which combined with the asymptotic expansions of Abelian integrals show that the cyclicity is three when the parameter belongs to $(-2,-\frac{8}{5})$.

Citation: Guilin Ji, Changjian Liu. The cyclicity of a class of quadratic reversible centers defining elliptic curves. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021299
##### References:

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##### References:
The phase portrait of system (2.1)
The real constant $\alpha_2$ and $\frac{W[\bar{F}_1,\bar{F}_3](x)}{W[\bar{F}_1,\bar{F}_2](x)}$ have at most two intersections
The phase portrait of system (1.5) with $b\in(-\infty,-2)\cup(-2,0]\cup[2,+\infty)$
The range of $(u,b).$
The value of $N_3(u,b)$
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