Hilberts16thProblemForQuadraticVectorFields 2003-10-31 16:26:32 2006-07-27 14:04:37 Conjecture Hilbert's sixteenth problem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 \cite{3} and also independently Chen Lan-Sun and Wang Ming-Shu \cite{4} showed an example of a quadratic vector field which has four limit cycles \textit{(i.e. $H(2)\geq 4$)}.\\ \textbf{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 $0<-\lambda\ll -\epsilon\ll- \delta\ll 1$. \cite{4}\\ \textbf{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 $0<\delta_2\ll\delta_1\ll 1$. \cite{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. \bibitem[ZTWZ]{4} Zhang Zhi-fen, Ding Tong-ren, Huang Wen-zoa, Dong Zhen-xi. Qualitative Theory of Differential Equations. American Mathematical Society, Providence, 1992. \end{thebibliography}