# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021314
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## The study on cyclicity of a class of cubic systems

 1 School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiaotong University, Shanghai 200240, China 2 College of Science, Zhongyuan University of Technology, Zhengzhou 450000, China

* Corresponding author: Jiang Yu

Received  July 2021 Revised  November 2021 Early access January 2022

Fund Project: The corresponding author is supported by the National Natural Science Foundations of China (11771282 and 11931016), Science and Technology Innovation Action Program of STCSM(20JC1413200), Innovation Program of Shanghai Municipal Education Commission (No. 2021-01-07-00-02-E00087). The first author is supported by the National Natural Science Foundations of China(11771282 and 12071285)

In this paper, we consider a class of cubic systems with polynomial perturbation of the degree at most $n$, and estimate the upper bound of the number of isolated zeros of its Abelian integral. Furthermore, we obtain the distributions of limit cycles bifurcated from a $Z_4$-equivariant system with $5$ centers.

Citation: Yuanyuan Chen, Jiang Yu. The study on cyclicity of a class of cubic systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021314
##### References:

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##### References:
Phase portraits of system (1.8) for $c>0$. $D_{1}^{+} = \{a>0, b^2 < 4ac\}, D_{2}^{+} = \{a<0, b>2a\}, D_{3}^{+} = \{a<0, b<2a\}, D_{4}^{+} = \{a<0, b^2>4ac\}, l_{1}^{+} = \{b = 2a, a<0\}, l_{2}^{+} = \{b^2 = 4ac, a>0\}$
Phase portraits of system (1.8) for $c<0$. $D_{1}^{-} = \{a>0,b\geq 2c\}, D_{2}^{-} = \{a<0,b^2>4ac\}, D_{3}^{-} = \{a<0,b^2<4ac,b>2c,b>2a\}, D_{4}^{-} = \{a<0,b\leq 2c,b\geq 2a\}, D_{5}^{-} = \{a<0,b\geq 2c,b\leq2a\},D_{6}^{-} = \{a<0,b<2c,b<2a\}, D_{7}^{-} = \{a>0,b<2c\}, l_{1}^{-} = \{b = 2a, a<0\}, l_{2}^{-} = \{b^2 = 4ac,a<0\}$
The distribution of 5 limit cycles of system (1.10)
$D_{3(2)}^{-}$
Phase portrait of unperturbed system of (1.10)
The distribution of limit cycles when $r(h) = \alpha$
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