LIMIT CYCLES BIFURCATED FROM A KIND OF PIECEWISE SMOOTH GENERALIZED ABEL EQUATION VIA THE FIRST ORDER MELNIKOV ANALYSIS

Cheng Wang; Fei Guan; Qianqian Zhao

doi:10.11948/20240338

2025 Volume 15 Issue 3

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Cheng Wang, Fei Guan, Qianqian Zhao. LIMIT CYCLES BIFURCATED FROM A KIND OF PIECEWISE SMOOTH GENERALIZED ABEL EQUATION VIA THE FIRST ORDER MELNIKOV ANALYSIS[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1695-1702. doi: 10.11948/20240338

Citation:

Cheng Wang, Fei Guan, Qianqian Zhao. LIMIT CYCLES BIFURCATED FROM A KIND OF PIECEWISE SMOOTH GENERALIZED ABEL EQUATION VIA THE FIRST ORDER MELNIKOV ANALYSIS[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1695-1702. doi: 10.11948/20240338

LIMIT CYCLES BIFURCATED FROM A KIND OF PIECEWISE SMOOTH GENERALIZED ABEL EQUATION VIA THE FIRST ORDER MELNIKOV ANALYSIS

1.
School of Applied Mathematics, Nanjing University of Finance and Economics, No. 3 Wenyuan Road, 210023 Nanjing, China
2.
College of Statistics and Mathematics, Hebei University of Economics and Business, No. 47 Xuefu Road, 050061 Shijiazhuang, China

Author Bio: Email: mathwc@126.com(C. Wang); Email: guanfei@hueb.edu.cn(F. Guan)
Corresponding author: Email: zqqyly@hueb.edu.cn(Q. Zhao)
Fund Project: The authors were supported by National Natural Science Foundation of China (12001265, 12301195), Natural Science Foundation of Jiangsu Province, China (BK20200829), Anhui Provincial Key Projects in Humanities and Social Sciences in Higher Education Institutions (SK2020A0475), Key Program of Hebei University of Economics and Business (2024ZD12) and Science Research Project of Hebei Education Department (QN2023168)

Abstract

Abstract

The study of the existence and distribution of limit cycles for generalized Abel equations comes from the famous Small-Pugh problem, which has been extended to non-smooth case. In this paper, we consider a kind of piecewise smooth generalized Abel equation with the separation line $ t=0 $. We are interested in its number of nontrivial limit cycles which are bifurcated from the periodic annulus of unperturbed equation. Under the first order Melnikov analysis, we show that the upper bound of this kind nontrivial limit cycles is $ 2(m+1) $ if $ p $ is odd, and $ m+1 $ if $ p $ is even. The upper bound in both cases can be reached separately.
- Piecewise smooth /
- generalized Abel equation /
- nontrivial limit cycles /
- bifurcation /
- Melnikov function
MSC: 34C07, 34C23, 34C25

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References

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