Topological and analytical classification of vector fields with only isochronous centres

Abstract

We study vector fields on the plane having only isochronous centres. The most familiar examples are isochronous vector fields, they are the real parts of complex polynomial vector fields on C having all their zeroes of centre type. We describe the number NðsÞ of topologically inequivalent isochronous (singular) foliations that can appear for degree s, up to orientation preserving homeomorphisms. For each s, there exists a real analytic variety I ðsÞ parametrizing the isochronous vector fields of degree s, the group of complex automorphisms of the plane AutðCÞ acts on it. Furthermore, if 2 # s # 7, then I ðsÞ is a non-singular real analytic variety of dimension s þ 3, and their number of connected components is bounded by 2NðsÞ. An explicit formula for the residues of the rational 1-form, canonically associated with a complex polynomial vector field with simple zeroes, is given. A collection of residues (i.e. periods) does not characterize an isochronous vector field, even up to complex automorphisms of C. An exact bound for the number of isochronous vector fields, up to AutðCÞ, having the same collection of residues (periods) is given. We develop several descriptions of the quotient space I ðsÞ=AutðCÞ using residues, weighted s-trees and singular flat Riemannian metrics associated with isochronous vector fields.

Dynamics of singular complex analytic vector fields with essential singularities I

“…Note that we mainly use the case z j = z k ; the pair of singularities z j , z k could be two poles (related to saddle connections, homoclinic, heteroclinic trajectories; see [13], [23]), a pole and an isolated essential singularity or even an isolated essential singularity of 1-order greater than 1, see for instance (8.9).…”

“…The point of view of differential equations (meromorphic vector fields): J. Gregor [26], [27], O. Hájek [31], [32], [33], N. A. Lukashevich [46], L. Brickman et al [14], M. Sabatini [59], J. Muciño-Raymundo et al [50], D. J. Needhan et al [52], E. P. Volokitin et al [70], A. Alvarez-Parrilla et al [4], A. Garijo et al [24], B. Branner et al [13], E. Frías-Armenta et al [23].…”

Configuration of Zeros of Isochronous Vector Fields of Degree 5

“…Theorem 1 (see [1,2]). Let X be a complex polynomial vector field on C of degree n ≥ 2 defined as in (1); then, the following statements are equivalent: (a) X has n isochronous centers (b) e zeros of X satisfy…”

“…We can associate to each isochronous vector field X a weighted n-tree in the following way. e n vertices correspond to the n zeros of X, and two vertices are connected with an edge if the basins of the corresponding centers are adjacent and the weights are the periods [1,3]. We know that each embedded n-tree (without weights) is realized by an isochronous vector field X and that if the phase portraits of two different isochronous vector fields are topologically equivalent, then they have the same embedded n-tree (see [3]).…”

“…A topological classification of the isochronous vector fields of degree 2 can be found in [1,[4][5][6]. In [1], the authors characterize the isochronous vector fields of degree n in terms of the shape of the configuration of zeros when n � 3 and 4, and they give partial results for n ≥ 5 when the zeros present some symmetries.…”