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
$\dot{x}=p(x,y)$ | $=$ | $\sum _{i+j=0}^{2}}{a}_{ij}{x}^{i}{y}^{j$ | ||
$\dot{y}=q(x,y)$ | $=$ | $\sum _{i+j=0}^{2}}{b}_{ij}{x}^{i}{y}^{j$ |
[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. $H\mathit{}\mathrm{(}\mathrm{2}\mathrm{)}\mathrm{\ge}\mathrm{4}$).
Example by Shi Songling:
The following system
$\dot{x}=$ | $\lambda x-y-10{x}^{2}+(5+\delta )xy+{y}^{2}$ | |||
$\dot{y}=$ | $x+{x}^{2}+(-25+8\u03f5-9\delta )xy$ |
has four limit cycles when $$. [ZTWZ]
Example by Chen Lan-sun and Wang Ming-Shu:
The following system
$\dot{x}=$ | $-y-{\delta}_{2}x-3{x}^{2}+(1-{\delta}_{1})xy+{y}^{2}$ | |||
$\dot{y}=$ | $x(1+{\displaystyle \frac{2}{3}}x-3y)$ |
has four limit cycles when $$. [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 |
---|---|
Canonical name | Hilberts16thProblemForQuadraticVectorFields |
Date of creation | 2013-03-22 14:03:35 |
Last modified on | 2013-03-22 14:03:35 |
Owner | Daume (40) |
Last modified by | Daume (40) |
Numerical id | 11 |
Author | Daume (40) |
Entry type | Conjecture |
Classification | msc 34C07 |