Rotated vector fields, global families of limit cycles and Hilbert’s 16th problem. (English) Zbl 0792.34028
Wiener, Joseph (ed.) et al., Ordinary and delay differential equations 1. Proceedings of the international conference on theory and applications of differential equations, held at the University of Texas-Pan American, Edinburg, TX, USA, on May 15-18, 1991. Harlow, Essex: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 272, 160-165 (1993).
The author proposes a program to prove that the number of limit cycles of a quadratic system is at most six. Using known facts about quadratic systems, the task is easily reduced to showing that no more than three limit cycles surround a given rest point. The idea of the program is to embed a quadratic system that is supposed to have at least four limit cycles surrounding a single rest point into a one parameter family of quadratic systems such that each element of the family has a multiplicity four limit cycle and such that the family of multiplicity four limit cycles terminates at a weak focus. This would violate Bautin’s Theorem and thus prove the desired result. As evidence for the possibility of such an embedding theorem, the author proves that a planar analytic system with \(k\) limit cycles surrounding a single rest point can be embedded in a family of analytic systems containing a one parameter family of multiplicity \(k\) limit cycles that terminate at a weak focus.
For the entire collection see [Zbl 0780.00043].
For the entire collection see [Zbl 0780.00043].
Reviewer: C.Chicone (Columbia)
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
