# Hilbert’s 16th problem for quadratic vector fields

Find a maximum natural number $H(2)$ and relative position of limit cycles of a vector field

 $\displaystyle\dot{x}=p(x,y)$ $\displaystyle=$ $\displaystyle\sum_{i+j=0}^{2}a_{ij}x^{i}y^{j}$ $\displaystyle\dot{y}=q(x,y)$ $\displaystyle=$ $\displaystyle\sum_{i+j=0}^{2}b_{ij}x^{i}y^{j}$

[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

 $\displaystyle\dot{x}=$ $\displaystyle\lambda x-y-10x^{2}+(5+\delta)xy+y^{2}$ $\displaystyle\dot{y}=$ $\displaystyle x+x^{2}+(-25+8\epsilon-9\delta)xy$

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

 $\displaystyle\dot{x}=$ $\displaystyle-y-\delta_{2}x-3x^{2}+(1-\delta_{1})xy+y^{2}$ $\displaystyle\dot{y}=$ $\displaystyle x(1+\frac{2}{3}x-3y)$

has four limit cycles when $0<\delta_{2}\ll\delta_{1}\ll 1$. [ZTWZ]

## References

• 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.
Title Hilbert’s 16th problem for quadratic vector fields Hilberts16thProblemForQuadraticVectorFields 2013-03-22 14:03:35 2013-03-22 14:03:35 Daume (40) Daume (40) 11 Daume (40) Conjecture msc 34C07