PlanetMath: Hilbert's 16th problem for quadratic vector fields

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (511)
Corrections (9)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Hilbert's 16th problem for quadratic vector fields

(Conjecture)

Find a maximum natural number and relative position of limit cycles of a vector field

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

[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

$\displaystyle \dot{x}=$		$\displaystyle -y -\delta_2x - 3x^2 + (1-\delta_1)xy + y^2$
$\displaystyle \dot{y}=$		$\displaystyle x(1+\frac{2}{3}x - 3y)$

has four limit cycles when $0<\delta_2\ll\delta_1\ll 1$ . [ZTWZ]

Bibliography

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.

"Hilbert's 16th problem for quadratic vector fields" is owned by Daume.

(view preamble)

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: Shi, number, finite, bound, vector field, limit cycles, natural number
There is 1 reference to this entry.

This is version 8 of Hilbert's 16th problem for quadratic vector fields, born on 2003-10-31, modified 2006-07-27.
Object id is 5415, canonical name is Hilberts16thProblemForQuadraticVectorFields.
Accessed 3906 times total.

Classification:

AMS MSC:

34C07 (Ordinary differential equations :: Qualitative theory :: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramif)

Pending Errata and Addenda

None.[ View all 4 ]

Discussion

forum policy

No messages.

Interact