The study on cyclicity of a class of cubic systems

Yuanyuan Chen; Jiang Yu

doi:10.3934/dcdsb.2021314

Article Contents

2022, Volume 27, Issue 11: 6233-6256. Doi: 10.3934/dcdsb.2021314

This issue Previous Article Strong pullback attractors for a nonclassical diffusion equation Next Article New insights to the Hide-Skeldon-Acheson dynamo

The study on cyclicity of a class of cubic systems

Yuanyuan Chen^1,2, and
Jiang Yu^1, ,

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
^* Corresponding author: Jiang Yu

Received: July 2021

Revised: November 2021

Early access: January 2022

Published: November 2022

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. 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\} $

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Figure 2. 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\} $

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Figure 3. The distribution of 5 limit cycles of system (1.10)

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Figure 4. $ D_{3(2)}^{-} $

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Figure 5. Phase portrait of unperturbed system of (1.10)

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Figure 6. The distribution of limit cycles when $ r(h) = \alpha $

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