The reduced Bautin index of planar vector fields.

*(English)*Zbl 0947.34013The determination of the number of small-amplitude limit cycles that bifurcate from a weak center of a planar polynomial system is a difficult problem that has only been solved for the case of quadratic systems (by N. N. Bautin). The basic approach to this problem starts with the displacement function \(S\) defined on a ray emanating from the weak center. If the center is taken to lie at the origin, then \(S\) is represented by a power series \(\sum_{k=0}^\infty a_k(z)X^k\) where \(X\) is the variable along the ray (usually taken to be a coordinate axis) and \(z\) is the vector variable of coefficients in the polynomial vector field. Because the ring of polynomials is Noetherian, there is a smallest integer \(d\), called the Bautin index, such that every series coefficient is in the ideal generated by \(\{a_0,a_1,\ldots,a_d\}\). The maximum number of small-amplitude limit cycles can generally be determined from the value of \(d\). For example, for the case of quadratic systems, \(d=7\) and the number of small-amplitude limit cycles is three. In the paper under review, the coefficients of the series expansion for the displacement function are proved to have a special growth property that implies, for each fixed \(z\), the existence of the positive constants \(\alpha\) and \(\beta\) such that the number of limit cycles can be bounded in a disk of the radius \(\alpha (1+|z|)^{-\beta}\). This bound is given in terms of the reduced Bautin index; that is, the smallest \(\bar d\) such that every series coefficient is in the integral closure of the ideal generated by \(\{a_0,a_1,\ldots,a_{\bar {d}}\}\). Although the results in the paper, which are obtained using a sophisticated blend of algebra and analysis, do not provide a bound for the number of small-amplitude limit cycles as a function of the degree of the polynomial vector field, they are a substantial contribution to the resolution of this intriguing problem.

Reviewer: C.Chicone (Columbia)

##### MSC:

34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |

37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

37C10 | Dynamics induced by flows and semiflows |

13E05 | Commutative Noetherian rings and modules |

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\textit{H. Hauser} et al., Duke Math. J. 100, No. 3, 425--445 (1999; Zbl 0947.34013)

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##### References:

[1] | N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type , American Math. Soc. Translation 1954 (1954), no. 100, 19. · Zbl 0059.08201 |

[2] | M. Briskin and Y. Yomdin, Algebraic families of analytic functions. I , J. Differential Equations 136 (1997), no. 2, 248-267. · Zbl 0886.34005 · doi:10.1006/jdeq.1996.3250 |

[3] | Azzeddine Fekak, Interprétation algébrique de l’exposant de Łojasiewicz , C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 4, 193-196. · Zbl 0706.14036 |

[4] | J.-P. Francoise and Y. Yomdin, Bernstein inequalities and applications to analytic geometry and differential equations , J. Funct. Anal. 146 (1997), no. 1, 185-205. · Zbl 0869.34008 · doi:10.1006/jfan.1996.3029 |

[5] | André Galligo, Théorème de division et stabilité en géométrie analytique locale , Ann. Inst. Fourier (Grenoble) 29 (1979), no. 2, vii, 107-184. · Zbl 0412.32011 · doi:10.5802/aif.745 · numdam:AIF_1979__29_2_107_0 · eudml:74406 |

[6] | M. Giusti and T. Mora, The complexity of Gröbner basis , preprint, 1994. |

[7] | Herwig Hauser and Gerd Müller, A rank theorem for analytic maps between power series spaces , Inst. Hautes Études Sci. Publ. Math. (1994), no. 80, 95-115 (1995). · Zbl 0831.58008 · doi:10.1007/BF02698897 · numdam:PMIHES_1994__80__95_0 · eudml:104102 |

[8] | M. Lejeune-Jalabert and B. Teissier, Clôture intégrale des idéaux et équisingularité , Sém. École Polytech., Publ. Institut Fourier, Grenoble, 1974. |

[9] | Joseph Lipman and Bernard Teissier, Pseudorational local rings and a theorem of Briançon-Skoda about integral closures of ideals , Michigan Math. J. 28 (1981), no. 1, 97-116. · Zbl 0464.13005 · doi:10.1307/mmj/1029002461 |

[10] | Maurice Mignotte, Mathématiques pour le calcul formel , Mathématiques. [Mathematics], Presses Universitaires de France, Paris, 1989. · Zbl 0679.12001 |

[11] | P. Monsky and G. Washnitzer, Formal cohomology. I , Ann. of Math. (2) 88 (1968), 181-217. JSTOR: · Zbl 0162.52504 · doi:10.2307/1970571 · links.jstor.org |

[12] | Y. Nesterenko and M. Waldschmidt, “On the approximation of the values of exponential function and logarithm by algebraic numbers“ (in Russian) , Diophantine Approximations: Proceedings of Papers Dedicated to the Memory of Prof. N. I. Feldman, Mat. Zapiski, vol. 2, Centre for Applied Research Under Mech.-Math. Faculty of MSU, Moscow, 1996, pp. 23-42. |

[13] | Pablo Solernó, Effective Łojasiewicz inequalities in semialgebraic geometry , Appl. Algebra Engrg. Comm. Comput. 2 (1991), no. 1, 2-14. · Zbl 0754.14035 · doi:10.1007/BF01810850 |

[14] | S. Yakovenko, A geometric proof of the Bautin theorem , Concerning the Hilbert 16th problem, Amer. Math. Soc. Transl. Ser. 2, vol. 165, Amer. Math. Soc., Providence, RI, 1995, pp. 203-219. · Zbl 0828.34026 |

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