New families of global cubic centers
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Springer Nature
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An equilibrium point p of a differential system in the plane R2 is a center if there exists a neighbourhood U of p such that U\{p} is filled with periodic orbits. A difficult classical problem in the qualitative theory of differential systems in the plane R2 is the problem of distinguishing between a focus and a center. A global center is a center p such that R2\{p} is filled with periodic orbits. Another difficult problem in the qualitative theory of differential systems in R2 is to distinguish inside a family of centers the ones which are global. Lloyd, Pearson and Romanovsky characterized when the origin of coordinates is a center for the family of cubic polynomial differential systems x˙=y-Cx2+B+2Dxy+Cy2+Px3+Gx2y-H+3Pxy2+Ky3,y˙=-x+Dx2+E+2Cxy-Dy2-Kx3-H+3Px2y-Gxy2+Py3. Here we characterize when the origin of this family of differential system is a global center.
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Sao Paulo Journal of Mathematical Sciences, v. 18, n. 2, p. 1454-1469, 2024.
