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The study on cyclicity of a class of cubic systems

  • * Corresponding author: Jiang Yu

    * Corresponding author: Jiang Yu

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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  • 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.

    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\} $

    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\} $

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

    Figure 4.  $ D_{3(2)}^{-} $

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

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

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