Bifurcations of limit cycles for a quintic Hamiltonian system with a double cuspidal loop
Bifurcations of limit cycles for a quintic Hamiltonian system with a double cuspidal loop
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DOI：
10.1016/j.camwa.2009.12.024
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发表时间：
2010-02
期刊：
影响因子：
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通讯作者：
R. Asheghi;H. Zangeneh
中科院分区：
文献类型：
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作者：
R. Asheghi;H. Zangeneh
在这篇文章中，我们考虑了在形式为ε（α+βx2+γx4）y y的小扰动下，五次哈密顿向量场XH=y x−x3（x2−1）y的双尖形环内的周期轨道可以分叉的极限环的个数，其中0<1，α，β，γ是真实的常数。利用相关交换积分的Picard-Fuchs方程，这些积分关于H的临界水平曲线的渐近展开式，以及由两个特殊积分之比定义的曲线的几何性质，我们证明了该分支中极限环个数的最小上界为2.
In this work we consider the number of limit cycles that can bifurcate from periodic orbits located inside a double cuspidal loop of the quintic Hamiltonian vector field XH=y∂∂x−x3(x2−1)∂∂y under small perturbations of the form ε(α+βx2+γx4)y∂∂y, where 0<∣ε∣≪1 and α,β,γ are real constants. Using Picard–Fuchs equations for related abelian integrals, asymptotic expansion of these integrals about critical level curves of H, and some geometric properties of the curves defined by ratios of two especial integrals, we show that the least upper bound for the number of limit cycles appeared in this bifurcation is two.