Find a maximum natural number $H(2)$ and relative position of limit cycles of a vector field \begin{eqnarray*} \dot{x} = p(x,y) &=&\sum_{i+j=0}^2 a_{ij}x^iy^j \\ \dot{y} = q(x,y) &=& \sum_{i+j=0}^2 b_{ij}x^iy^j \end{eqnarray*}[DRR].
As of now neither part of the problem (i.e. the bound and the positions of the limit cycles) are solved. Although R. Bamòn in 1986 showed [BR] that a quadratic vector field has finite number of limit cycles. In 1980 Shi Songling [SS] and also independently Chen Lan-Sun and Wang Ming-Shu [ZTWZ] showed an example of a quadratic vector field which has four limit cycles (i.e. $H(2)\geq 4$ ).

Example by Shi Songling:
The following system \begin{eqnarray*} \dot{x}=& \lambda x - y - 10x^2 + (5+\delta)xy + y^2 \\ \dot{y}=& x + x^2 + (-25 + 8\epsilon - 9\delta)xy \end{eqnarray*}has four limit cycles when $0<-\lambda\ll -\epsilon\ll- \delta\ll 1$ . [ZTWZ]

Example by Chen Lan-sun and Wang Ming-Shu:
The following system \begin{eqnarray*} \dot{x}=& -y -\delta_2x - 3x^2 + (1-\delta_1)xy + y^2\\ \dot{y}=& x(1+\frac{2}{3}x - 3y) \end{eqnarray*}has four limit cycles when $0<\delta_2\ll\delta_1\ll 1$ . [ZTWZ]

Bibliography

DRR

Dumortier, F., Roussarie, R., Rousseau, C.: Hilbert's 16th Problem for Quadratic Vector Fields. Journal of Differential Equations 110, 86-133, 1994.

BR

R. Bamòn: Quadratic vector fields in the plane have a finite number of limit cycles, Publ. I.H.E.S. 64 (1986), 111-142.

SS

Shi Songling, A concrete example of the existence of four limit cycles for plane quadratic systems, Scientia Sinica 23 (1980), 154-158.

ZTWZ

Zhang Zhi-fen, Ding Tong-ren, Huang Wen-zoa, Dong Zhen-xi. Qualitative Theory of Differential Equations. American Mathematical Society, Providence, 1992.