ABELIAN INTEGRALS FOR A KIND OF QUADRATIC REVERSIBLE CENTERS OF GENUS ONE (<i>R</i>7)

Lijun Hong; Bin Wang; Xiaochun Hong

doi:10.11948/20210487

2022 Volume 12 Issue 4

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Lijun Hong, Bin Wang, Xiaochun Hong. ABELIAN INTEGRALS FOR A KIND OF QUADRATIC REVERSIBLE CENTERS OF GENUS ONE (R7)[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1624-1635. doi: 10.11948/20210487

Citation:

Lijun Hong, Bin Wang, Xiaochun Hong. ABELIAN INTEGRALS FOR A KIND OF QUADRATIC REVERSIBLE CENTERS OF GENUS ONE (R7)[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1624-1635. doi: 10.11948/20210487

ABELIAN INTEGRALS FOR A KIND OF QUADRATIC REVERSIBLE CENTERS OF GENUS ONE (R7)

1.
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
2.
School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China

Corresponding author: Email address: xchong@ynufe.edu.cn(X. Hong)
Fund Project: This work was supported by the National Natural Science Foundation of China (No. 11761075)

Abstract

Abstract

For the quadratic reversible centers of genus one $ (r7) $, its all periodic orbits are quartic curves. Using the method of Picard-Fuchs equation and Riccati equation, we study that the upper bound of the number of zeros for Abelian integrals of system $ (r7) $ under arbitrary polynomial perturbations of degree $ n $, and obtain that the upper bound of the number is $ 45n-72 $ when $ n\geq 2 $, $ 5 $ when $ n=1 $, and $ 0 $ when $ n=0 $, which depends linearly on $ n $.
- Abelian integral /
- limit cycle /
- quadratic reversible center /
- weakened Hilbert's 16th problem
MSC: 34C07, 34C08, 37G15

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References

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