Normal Form for a Class of Three-Dimensional Systems with Free-Divergence Principal Part

On the integrability problem for the Hopf-zero singularity and its relation with the inverse Jacobi multiplier

“…In this section, we collect results from [3] that will be used along this paper. Among them, the main result is Theorem 2.2, where we present the orbital normal form for system (1.1).…”

“…Next, we present the orbital normal form for system (1.1) obtained in [3]. It determines how much system (1.1) can be simplified by means of an infinite sequence of time-reparametrizations and near-identity coordinate transformations.…”

“…In such a way, we can achieve (∇I • F) k = R c k ∈ Cor (ℓ k ) , a complementary subspace to Range(ℓ k ). In [3], it is obtained that, if k is even, then Cor(ℓ k ) = Span z k/2 and Ker (ℓ k ) = Span h k/2 , whereas if k is odd, then Cor(ℓ 2k 1 +1 ) = Ker (ℓ k ) = {0}.…”

“…This is still a difficult problem and there are few known satisfactory methods to solve it. In the present paper, we use the orbital normal form for system (1.1) obtained in [3] to establish necessary conditions for the existence of first integrals for the nondegenerate Hopf-zero singularity (1.1). Moreover, we analyze the relation between the existence of first integrals and of inverse Jacobi multipliers.…”

“…This paper is organized as follows. In Section 2 we include definitions and results about quasi-homogeneous vector fields and the nondegenerate Hopf-zero orbital normal form, that we use in this work (their proofs can be found in [3]). In Section 3, the orbital normal form is used to obtain some results about the analytic integrability of this singularity.…”