Qualitative Theory of Dynamical Systems

, Volume 18, Issue 3, pp 947–967 | Cite as

On the Number of Zeros of Abelian Integral for a Class of Cubic Hamilton Systems with the Phase Portrait “Butterfly”

  • Jihua Yang
  • Shiyou Sui
  • Liqin ZhaoEmail author


The present paper is devoted to study the number of zeros of Abelian integral for the near-Hamilton system
$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{x} = 2y(bx^2+2cy^2)+\varepsilon f(x,y),\\ \dot{y} = 2x(1-2ax^2-by^2)+\varepsilon g(x,y), \end{array}\right. } \end{aligned}$$
where \(a,b,c\in \mathbb {R}\), \(b<0\), \(c>0\), \(b^2<4ac\), \(0<|\varepsilon |\ll 1\), f(xy) and g(xy) are polynomials in (xy) of degree n. The generators of the corresponding Abelian integral satisfy three different Picard–Fuchs equations. We obtain an upper bound of the number of isolated zeros of the Abelian integral.


Hamilton system Abelian integral Weakened Hilbert’s 16th problem Picard–Fuchs equation Chebyshev space 



Funding was provided by National Natural Science Foundation of China (Grant Nos. 11701306 and 11671040) and Construction of First-class Disciplines of Higher Education of Ningxia (pedagogy) (Grant Nos. NXYLXK2017B11).


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Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceNingxia Normal UniversityGuyuanChina
  2. 2.Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical SciencesBeijing Normal UniversityBeijingChina

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