Topological classification of polynomial complex differential equations with all the critical points of centre type

Abstract

In this paper we study the global phase portrait of complex polynomial differential equations of degree n of the formż = f (z), having all their critical points of center type. We give the exact number of topologically different phase portraits on the Poincaré disk when n ≤ 6 and, in the remaining cases, an upper bound for this number which only depends on n.

Classification of complex polynomial vector fields in one complex variable

“…Although no limit cycles exist for holomorphic vector fields, an important strategy when dealing with the Hilbert's 16th problem is to study perturbations of holomorphic vector fields with centres [1,12].…”

Topological and analytical classification of vector fields with only isochronous centres

“…(i) looking at the classes of embeddings up to homeomorphisms of the plane, HomeoðCÞ, as was studied by Á lvarez et al [2] and Rong [24], or (ii) looking as plane s-trees, i.e. classes of embeddings up to orientation preserving homeomorphisms of the plane, HomeoðCÞ þ , see diagram (15).…”

“…The proof of the right equality in (15) uses arguments in the phase portraits as in [2], we leave the details to the reader. A virtue of the concept of plane s-tree is its relation with the number of connected components in I ðsÞ and I ðsÞ=AutðCÞ, as we see in Theorem 6.1.3.…”

“…We answer a question in [2] describing the topological classification of isochronous vector fields X [ I ðsÞ. In order to avoid confusions, we make explicit three notions of equivalence.…”

“…They are good prospects to study the birth of limit cycles under perturbation of centres, this is the infinitesimal Hilbert's 16 problem [2,14]. In the topological classification of plane polynomial vector fields, the phase portraits of polynomial vector fields with only isochronous centres are between the simplest (see [1,4] or [23]) for the quadratic case.…”