Abstract
In this paper, we give the algebraic conditions that a configuration of 5 points in the plane must satisfy in order to be the configuration of zeros of a polynomial isochronous vector field. We use the obtained results to analyze configurations having some of its zeros satisfying some particular geometric conditions.
Highlights
We start defining an isochronous vector field, and we express its general associated 1-form, with its respective residues.An isochronous vector field X is as a complex polynomial vector field on C whose zeros are all isochronous centers
We give the algebraic conditions that a configuration of 5 points in the plane must satisfy in order to be the configuration of zeros of a polynomial isochronous vector field
(a) For each n ≥ 3, if the zeros p1, p2, . . . , pn are in a line, X is isochronous and his phase portrait has the line topology (b) For each n ≥ 4, if the zeros p2, p3, . . . , pn are at the vertices of a regular polygon and p1 is at its center, X is isochronous and his phase portrait has the star topology (c) For each n ≥ 4, there exist isochronous vector fields with the zeros p1, . . . , pn− 2 in a line and the zeros pn− 1 and pn in new line orthogonal to the previous one In addition, for n 5, the following statements hold
Summary
We start defining an isochronous vector field, and we express its general associated 1-form, with its respective residues.An isochronous vector field X is as a complex polynomial vector field on C whose zeros are all isochronous centers. Let X be a complex polynomial vector field on C of degree n ≥ 2 defined as in (1); the following statements are equivalent: 2 (a) X has n isochronous centers (b) e zeros of X satisfy p1 · · · pj − pj · · · pj − Erefore, all the solutions (x3, y3, x5, y5) of equation g72 0 that do not satisfy conditions Ki with i 1, 2, and 3 provide isochronous configurations with (x4, y4) (x40, y40).
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