On a Class of Rigid Polynomial Centers

期刊论文
发表日期：

2025年8月23日

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Jaume Llibre,
Claudia Valls

摘要

When all the orbits of a planar differential system in a punctured neighborhood of a singular point are periodic, we say that the singular point is a center. The topological annulus formed only by all periodic orbits surrounding a center is called the period annulus of the center. When the periods of the periodic orbits of a period annulus are equal, the center is called isochronous. Moreover, an isochronous center is rigid (or uniform) if the angular velocity of its periodic orbits is constant. It is well-known that the rigid polynomial centers at the origin of coordinates can be written as \(x'=-y + x P(x,y)\) , \(y' =x + y P(x,y)\) , where P(x, y) is a polynomial. However, not all these systems have a center at the origin of coordinates, the origin can also be a focus. In this paper we characterize the rigid centers when P(x, y) is a homogeneous polynomial of degree \(n \ge 1\) . The phase portraits of the rigid polynomial homogeneous centers where P(x, y) has degrees 1,2, or 3 are well-known. Here we provide the phase portraits of the rigid polynomial homogeneous centers when P(x, y) has degree four.