Hilbert’s 16th problem for quadratic vector fields
Find a maximum natural number and relative position of limit cycles of a vector field
[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. ).
Example by Shi Songling:
The following system
has four limit cycles when . [ZTWZ]
Example by Chen Lan-sun and Wang Ming-Shu:
The following system
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.