TEN LIMIT CYCLES NEAR A CUBIC HOMOCLINIC LOOP WITH A NILPOTENT CUSP

Yun Tian; Didi Ma

doi:10.11948/20260103

2026 Volume 16 Issue 5

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Yun Tian, Didi Ma. TEN LIMIT CYCLES NEAR A CUBIC HOMOCLINIC LOOP WITH A NILPOTENT CUSP[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2569-2583. doi: 10.11948/20260103

Citation:

Yun Tian, Didi Ma. TEN LIMIT CYCLES NEAR A CUBIC HOMOCLINIC LOOP WITH A NILPOTENT CUSP[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2569-2583. doi: 10.11948/20260103

TEN LIMIT CYCLES NEAR A CUBIC HOMOCLINIC LOOP WITH A NILPOTENT CUSP

Yun Tian^1, ,,
Didi Ma^1,

1.
Department of Mathematics, Shanghai Normal University, 100 Guilin Road, 200234 Shanghai, China

The first draft of this paper has been posted on arXiv.org, No. 2310.05234.
Author Bio: Email: 1750796475@qq.com(D. Ma)
Corresponding author: Email: ytian22@shnu.edu.cn(Y. Tian)
Fund Project: The authors were supported by National Natural Science Foundation of China (12371175 and 11871042)

Abstract

Abstract

In this paper, we study the bifurcation of limit cycles near a homoclinic cuspidal loop in a planar cubic near-Hamiltonian system by high-order Melnikov functions. We combine the algebraic structure of Abelian integrals with Picard-Fuchs equations for computing the corresponding asymptotic expansion of Melnikov functions near the cuspidal loop. Using this system as an example, we show that planar cubic systems can have ten limit cycles bifurcating near a homoclinic loop, which is a new lower bound for the number of limit cycles produced by homoclinic bifurcation in cubic systems.
- Homoclinic bifurcation /
- limit cycle /
- high-order Melnikov function
MSC: 34C07

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References

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