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'Hilbert's 16th problem for quadratic vector fields'
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| Title of object: |
Hilbert's 16th problem for quadratic vector fields |
| Canonical Name: |
Hilberts16thProblemForQuadraticVectorFields |
| Type: |
Conjecture |
| Created on: |
2003-10-31 16:26:32 |
| Modified on: |
2003-11-10 16:38:28 |
| Classification: |
msc:34C07 |
Preamble:
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Content:
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*}
\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}
\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} |
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