Cyclicity of nilpotent centers with minimum Andreev number

Abstract

We consider polynomial families of real planar vector fields for which the origin is a monodromic nilpotent singularity having minimum Andree's number. There the centers are characterized by the existence of a formal inverse integrating factor. For such families we give, under some assumptions, global bounds on the maximum number of limit cycles that can bifurcate from the singularity under perturbations within the family.

Center cyclicity for some nilpotent singularities including the ℤ2-equivariant class

“…The strong requirement I odd = {0}, which implies in particular the weaker condition I = I even in Corollary 10(iii), is always reached (choosing adequate freedoms τ j (λ)) in case of having the minimum Andreev number n = 2 and β = 1, see [24]. In the next theorem we will see how to detect other cases with I odd = {0} under some symmetry conditions on the Andreev canonical form (2).…”

“…The ideas involved in these proof are based on the adaptation of known results to our framework, which is rather straightforward under the assumption I = I even which implies B = I taking into account our Theorem 8. Besides using general facts established in books such as [42], [43] or [34], these proofs mainly apply constructions from [29], [23] and [24], this last reference strongly based on the work [28]. Theorem 22 ultimately relies on results in [13].…”

“…Essentially we need to replace the concepts of (i) nondegenerate planar monodromic singularity or Hopf singularity in R 3 ; (ii) formal first integral; (iii) focal values; and (iv) admissible parameter in the linear part of the vector field in R 3 lying in R * by (i) monodromic nilpotent singularity with I = I even ; (ii) formal inverse integrating factor; (iii) integrating focal values; (iv) when β = n−1 the admissible parameter in the (1, n)-quasihomogeneous terms of the vector field lying in Ω n . This approach has already been used successfully in [24] when n = 2.…”

“…Here we interpret the elements of R(ω) as just formal expressions so R(ω) is a field, hence R(ω)[ν] is a Noetherian ring. Taking advantage of these algebraic structures we can use some tools of algebraic geometry to construct bounds on Cyc(X , 0) where the analyzed families strictly contain the Z 2 -equivariant family (as follows by Theorem 12 and Proposition 13) and also the specific case n = 2, see [24]. A consequence of the work is the existence of a center variety V C ⊂ R p associated to the origin of the studied families (16) which is an algebraic variety in the parameter space having the property that the center corresponds with the parameters that lie in V C ∩ E. We note that V C is the variety of the polynomial Bautin ideal B = v j : j ∈ N only when β = n − 1 and is the variety of certain polynomial ideal (called G is this work) associated to B otherwise.…”

“…This is why only few general results are known on the nilpotent center cyclicity problem. The techniques introduced in this work generalize to those given in [23], [24] and [29].…”