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The existential Hilbert 16-th problem and an estimate for cyclicity of elementary polycycles. (English) Zbl 1087.34013

Summary: An important chapter of the theory of planar vector fields deals with finiteness problems for the number of limit cycles. Global finiteness results usually rely on local finiteness results, namely problems of estimating the finite cyclicity of polycycles. There are only a few results of finite cyclicity for polycycles with nonelementary singular points [F. Dumortier, R. Roussarie and J. Sotomayor, Nonlinearity 10, No. 6, 1369–1408 (1997; Zbl 0908.58043); H. Zhu and C. Rousseau, J. Differ. Equations 178, No. 2, 325–436 (2002; Zbl 1012.34028)], and general results are completely out of reach because of a lack of good normal forms for the families unfolding the vector fields in the neighborhood of the singular points. On the other hand this paper treats the general case of generic elementary polycycles: all singular points on the polycycle have at least one nonzero eigenvalue.
The paper gives a complete and new proof of one of the major theorems of the subject which was first proved by Y. S. Il’yashenko and S. Yakovenko [Yu. Il’yashenko (ed.) et al., Concerning the Hilbert 16th problem. Providence, RI: American Mathematical Society. Transl., Ser. 2, A. Math. Soc. 165(23), 1–19 (1995; Zbl 0828.34020)] (hereafter [IY]): “Any nontrivial elementary polycycle occurring in a generic \(k\)-parameter family of \(C^\infty\) vector fields has finite cyclicity.” A stronger form of the theorem is proved with the first explicit estimate in terms of \(k\) alone. Indeed, let \(E(k)\) be the maximal cyclicity of a nontrivial elementary polycycle occurring in a generic \(k\)-parameter family. Then the main theorem of the paper is the following: “For any \(k\in\mathbb{N}\), \(E(k)\leq 2^{25k^2}\).” The paper gives an excellent survey of the main avenues of research in this area. Not only is the result of the paper important, so is the method of proof. The paper refines the Khovanskiĭ procedure and shows how to give an explicit upper bound on the geometric multiplicity of a “chain-map” which is defined as the germ of a map \(P\circ F\), where \(F:\mathbb{R}^n\to\mathbb{R}^N\) is a generic map and \(P:\mathbb{R}^N\to \mathbb{R}^n\) is a polynomial map. The general methods introduced in the paper are likely to have a lot of applications.
The proof of the main theorem is quite sophisticated and shows significant improvements to the proof of [IY]. Like the proof of [IY], it relies on the method of Khovanskiĭ. Indeed, to bound the cyclicity of a polycycle of a given vector field inside a generic \(k\)-parameter family, one must bound the number of fixed-points of the Poincaré return map in the neighborhood of the polycycle for neighboring values of the parameters. To calculate the first return map one introduces integrable \(C^n\) normal forms for the family of vector fields in the neighborhood of the singular points, so as to calculate the Dulac maps, which are the transition maps in the neighborhood of the singular points. For a polycycle with \(m\) singular points, the fixed-points of the Poincaré return map are given by the solutions of a system of \(2m\) equations. Half of these equations are of the form \(y_i=\Delta_i (x_i)\) where \(\Delta_i\) is the Dulac map near the \(i\)th singular point. The first step of the Khovanskiĭ procedure consists in replacing these equations, which are highly transcendental, by polynomial Pfaffian equations (polynomial 1-forms): the graphs of the Dulac maps are separating solutions of these Pfaffian equations. We then get a mixed functional-Pfaffian system, the only functions appearing in the system being generic \(C^\infty\)-functions. The second step of the Khovanskiĭ procedure is to get rid of the Pfaffian equations. Indeed, the number of solutions of a mixed functional-Pfaffian system is bounded by the sum of the number of solutions of two new systems where one Pfaffian equation has been replaced with a functional equation. One finally ends with finding a bound to the number of solutions of several functional systems which we will call auxiliary systems. The whole process must be carried out in a constructive and careful way in order to obtain an explicit bound. For instance, the degree of the polynomial Pfaffian equations satisfied by the Dulac maps is bounded in terms of \(k\), as the codimension of the polycycle is at most \(k\). Also, a single Dulac map may be replaced by a system of three Pfaffian equations if one wants to keep control of the degree of the Pfaffian equations. Finally, the genericity conditions satisfied by the regular transitions between the singular points can be read on a \((6k+1)\)-jet. All systems for which we must bound the number of solutions are finally put in the form of a “polynomial chain map”, which is a composition of a known polynomial vector function \(P\) with a generic vector-valued \(C^\infty\) function.
The third step of the Khovanskiĭ procedure consists in limiting ourselves to counting the number of regular solutions of the systems. A subtle point of the proof is to construct a stratification of the jet-space adapted to the problem, an \({\mathcal A}_P\)-stratification, so that if an adequate jet of the functions is transversal to all strata then we can replace the jets in the systems by their linearizations. In general, \({\mathcal A}_P\) stratifications do not exist. The very clever trick, which is a real improvement on [IY], is to show that an \({\mathcal A}_P\)-stratification exists if we limit ourselves to a very thin cone.

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
37C10 Dynamics induced by flows and semiflows
37G10 Bifurcations of singular points in dynamical systems
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