Hilbert’s 16th problem for quadratic vector fields

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

x˙=p(x,y) = i+j=02aijxiyj
y˙=q(x,y) = i+j=02bijxiyj

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)4).

Example by Shi Songling:
The following system

x˙= λx-y-10x2+(5+δ)xy+y2
y˙= x+x2+(-25+8ϵ-9δ)xy

has four limit cycles when 0<-λ-ϵ-δ1. [ZTWZ]

Example by Chen Lan-sun and Wang Ming-Shu:
The following system

x˙= -y-δ2x-3x2+(1-δ1)xy+y2
y˙= x(1+23x-3y)

has four limit cycles when 0<δ2δ11. [ZTWZ]


  • DRR Dumortier, F., Roussarie, R., Rousseau, C.: Hilbert’s 16th Problem for Quadratic Vector Fields. Journal of Differential EquationsMathworldPlanetmath 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