| Version 3 |
Version 2 |
| Find a maximum natural number $H(2)$ and relative position of limit cycles of a vector field |
Find a maximum natural number $H(2)$ and relative position of limit cycles of a vector field |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \dot{x} = p(x,y) &=&\sum_{i+j=0}^2 a_{ij}x^iy^j \\ |
\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 |
\dot{y} = q(x,y) &=& \sum_{i+j=0}^2 b_{ij}x^iy^j |
| \end{eqnarray*} |
\end{eqnarray*} |
| \cite{1}\\ |
\cite{1} |
| As of now neither part of the problem \textit{(i.e. the bound and the positions of the limit cycles)} are solved. Although R. Bam\'on in 1986 showed \cite{2} that a quadratic vector field has finite number of limit cycles. In 1980 Shi Songling showed \cite{3} an example of a quadratic vector field which has four limit cycles \textit{(i.e. $H(2)\geq 4$)}. |
|
| \begin{thebibliography}{1} |
\begin{thebibliography}{1} |
| \bibitem[DRR]{1} Dumortier, F., Roussarie, R., Rousseau, C.: Hilbert's 16th Problem for Quadratic Vector Fields. Journal of Differential Equations 110, 86-133, 1994. |
\bibitem[DRR]{1} Dumortier, F., Roussarie, R., Rousseau, C.: Hilbert's 16th Problem for Quadratic Vector Fields. Journal of Differential Equations 110, 86-133, 1994. |
| \bibitem[BR]{2} R. Bam\'on: Quadratic vector fields in the plane have a finite number of limit cycles, Publ. I.H.E.S. 64 (1986), 111-142. |
|
| \bibitem[SS]{3} Shi Songling, A concrete example of the existence of four limit cycles for plane quadratic systems, Scientia Sinica 23 (1980), 154-158. |
|
| \end{thebibliography} |
\end{thebibliography} |
