Nilpotence of orbits under monodromy and the length of Melnikov functions☆
Section snippets
Introduction, main results and conjectures
This work is motivated by the 16th Hilbert’s problem or rather its infinitesimal version. As it is known, the second part of Hilbert’s 16-th problem asks for an upper bound in terms of the degree for the number of real limit cycles, i.e. isolated periodic orbits of polynomial vector fields in the plane. The problem is far from been solved and the existence of such a number is open even for quadratic vector fields.
Arnold formulated the infinitesimal Hilbert’s problem, which asks for a bound on
Nilpotence class and derivative length
Definition 2.1 Given a group , let be its lower central sequence: If there exists such that , we say that the group is nilpotent and define its nilpotence class as where is the identity element in . Similarly, the upper central sequence is If there exists such that , we say that the group is solvable and define its derived length as
Note that
This gives and in particular any
Germs of diffeomorphisms
Given a germ of diffeomorphism , we say that it is parabolic if it is of the form . If is not the identity, then , with . We call the level of . Let denote the subgroup of parabolic germs.
The general approach given in 1.4 is based on the following well-known facts about the solvability of the group of parabolic germs .
Lemma 3.1 Let and . Then .Proposition 6.11, [10]
Proposition 3.2 Let be a finitelyLemma 6.13 [10]
Proof of Theorem 1.5
Proof In the case (1), of product of lines in generic position, it has been proved in [8] that . Therefore, is abelian, and . Hence, the orbit depth of the real cycle, as well as the length of the first non-zero Melnikov function for any deformation, is bounded by 2, by [4]. To prove that is non-solvable for cases (2) and (3) we follow the same strategy. In both cases, by an affine change of coordinates we can assume that the Hamiltonian is given by , where
Types of integrability of the deformations
We will study the Godbillon–Vey sequence for the foliation with and , where , , are polynomials in , and , are linear factors different from and . For the parallelogram, and define parallel lines, while for the trapezoid they do not. By explicit computation of Godbillon–Vey sequences we will show that the foliation is Liouville integrable, while is Riccati integrable. Moreover,
Proof of Theorem 1.7
Proof We consider the deformation , where , , and define Denote . Then , and since , where is the partial derivative of with respect to : , we have Thus, we define . It satisfies the equation . Now we consider . Then, again, writing , we have . So, we can take . It satisfies the equation . Continuing with
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
We would like to thank Frank Loray for usefull discussions which helped clarify some points in this article.
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This work was supported by Israel Science Foundation grant 1167/17, Papiit (Dgapa UNAM), Mexico IN110520, Croatian Science Foundation (HRZZ) Grant Nos. 2285, UIP 2017-05-1020, PZS-2019-02-3055 from Research Cooperability funded by the European Social Fund .