A note on finite cyclicity property and Hilbert’s 16th problem.

*(English)*Zbl 0676.58046
Dynamical systems, Proc. Symp., Valparaiso/Chile 1986, Lect. Notes Math. 1331, 161-168 (1988).

[For the entire collection see Zbl 0647.00005.]

This paper is a contribution to the exciting recent work on Hilbert’s 16th problem on the number of limit cycles of a polynomial vector field. Hilbert asked if, for each n, there exists a bound H(n) for the number of limit cycles of any polynomial vector field of degree n. As a natural first step in this direction one asks if each polynomial vector field has at most a finite number of limit cycles: Dulac’s problem. Recently J. Ecalle, J. Martinet, R. Moussu and J. P. Ramis [C. R. Acad. Sci., Paris, Sér. I 304, 375-377; 431-434 (1987; Zbl 0615.58011)] announced a complete proof of this problem. Secondly, assuming the finiteness problem settled, one asks if a bound depending only on the degree exists.

In the paper under review a program is given for obtaining the existence of such a bound. This work is in the spirit of the classic paper of N. N. Bautin [Am. Math. Soc., Transl. 100, 1-19 (1954); translation from Mat. Sb., Nov. Ser. 30(72), 181-196 (1952; Zbl 0046.094)] where the notion of finite cyclicity for bifurcations was first defined and where it was proved that a center or weak focus of a quadratic system has cyclicity at most three.

The author gives the appropriate definition of cyclicity in order to include the bifurcations of limit cycles from separatrix cycles. He conjectures that this cyclicity is finite and proves that his conjecture can be proved by a local analysis. In fact, this analysis has been carried out by J. P. Françoise and C. C. Pugh [J. Differ. Equations 65, 139-157 (1986; Zbl 0602.34019)] for elliptic points and periodic orbits and by the author for hyperbolic loops [Bol. Soc. Bras. Mat. 17, No.1, 67-101 (1986; Zbl 0628.34032)] and for cusps in recent, as yet unpublished, work. These results should be seriously considered by anyone interested in Hilbert’s problem.

This paper is a contribution to the exciting recent work on Hilbert’s 16th problem on the number of limit cycles of a polynomial vector field. Hilbert asked if, for each n, there exists a bound H(n) for the number of limit cycles of any polynomial vector field of degree n. As a natural first step in this direction one asks if each polynomial vector field has at most a finite number of limit cycles: Dulac’s problem. Recently J. Ecalle, J. Martinet, R. Moussu and J. P. Ramis [C. R. Acad. Sci., Paris, Sér. I 304, 375-377; 431-434 (1987; Zbl 0615.58011)] announced a complete proof of this problem. Secondly, assuming the finiteness problem settled, one asks if a bound depending only on the degree exists.

In the paper under review a program is given for obtaining the existence of such a bound. This work is in the spirit of the classic paper of N. N. Bautin [Am. Math. Soc., Transl. 100, 1-19 (1954); translation from Mat. Sb., Nov. Ser. 30(72), 181-196 (1952; Zbl 0046.094)] where the notion of finite cyclicity for bifurcations was first defined and where it was proved that a center or weak focus of a quadratic system has cyclicity at most three.

The author gives the appropriate definition of cyclicity in order to include the bifurcations of limit cycles from separatrix cycles. He conjectures that this cyclicity is finite and proves that his conjecture can be proved by a local analysis. In fact, this analysis has been carried out by J. P. Françoise and C. C. Pugh [J. Differ. Equations 65, 139-157 (1986; Zbl 0602.34019)] for elliptic points and periodic orbits and by the author for hyperbolic loops [Bol. Soc. Bras. Mat. 17, No.1, 67-101 (1986; Zbl 0628.34032)] and for cusps in recent, as yet unpublished, work. These results should be seriously considered by anyone interested in Hilbert’s problem.

Reviewer: C.Chicone