NEW UPPER BOUNDS ON THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR THREE PERTURBED QUADRATIC REVERSIBLE SYSTEMS

Xiaochun Hong; Jing Chang; Lijun Hong; Ronghua Cheng; Wei Lu

doi:10.11948/20260150

2027 Volume 17 Issue 2

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Xiaochun Hong, Jing Chang, Lijun Hong, Ronghua Cheng, Wei Lu. NEW UPPER BOUNDS ON THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR THREE PERTURBED QUADRATIC REVERSIBLE SYSTEMS[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1253-1269. doi: 10.11948/20260150

Citation:

Xiaochun Hong, Jing Chang, Lijun Hong, Ronghua Cheng, Wei Lu. NEW UPPER BOUNDS ON THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR THREE PERTURBED QUADRATIC REVERSIBLE SYSTEMS[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1253-1269. doi: 10.11948/20260150

NEW UPPER BOUNDS ON THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR THREE PERTURBED QUADRATIC REVERSIBLE SYSTEMS

1.
School of Data Science, Zhuhai College of Science and Technology, Zhuhai 519040, China
2.
School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China
3.
School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China

Author Bio: Email: xchong@zcst.edu.cn(X. Hong); Email: 9909513@qq.com(J. Chang); Email: honglj6@mail.sysu.edu.cn(L. Hong); Email: luwei@zcst.edu.cn(W. Lu)
Corresponding author: Email: kmchrh@126.com(R. Cheng)
Fund Project: The authors were supported by National Natural Science Foundation of China (Grant No. 11761075), the Tri-level Talents Development Project of Zhuhai College of Science and Technology, the China Postdoctoral Science Foundation (Grant No. 2026M793425), the Key Research Platforms and Projects of Ordinary Universities under the Eduction Department of Guangdong Provincial (Grant No. 2023ZDZX1049), the Fund Project of Yunnan University of Finance and Economics (Grant No. 2025H36)

Abstract

Abstract

For the quadratic reversible systems $r12$, $r16$ and $r18$, when they are perturbed by arbitrary polynomial of degree $n$, we explore the new upper bounds on the number of zeros for their Abelian integrals using the methods of Riccati equation and Picard-Fuchs equation. We consider both the highest degree and the lowest degree of the related polynomials, and more importantly, we notice their law between the terms, and take into account the range of values for variable. Some laws of related polynomials of the systems have been found, therefore, we obtain some new upper bounds, which are better than those obtained before.
- Abelian integral /
- limit cycle /
- upper bound /
- quadratic reversible system /
- weakened Hilbert's 16th problem
MSC: 34C07, 34C08, 37G15

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References

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