Global phase portraits of Kukles differential systems with homogeneous polynomial nonlinearities of degree 5 having a center. (English) Zbl 1360.34071

Llibre, Jaume; da Silva, Maurício Fronza

Topol. Methods Nonlinear Anal. 48, No. 1, 257-282 (2016).

Summary: We provide 22 different global phase portraits in the Poincaré disk of all centers of the so-called Kukies polynomial differential systems of the form \[ \dot x= -y,\;\dot y= x +Q_5(x, y), \] where \(Q_5\) is a real homogeneous polynomial of degree 5 defined in \(\mathbb{R}^2\).

MSC:

34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C07	Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C25	Periodic solutions to ordinary differential equations

Keywords:

centers; Kukles; polynomial vector fields; phase portrait; Poincaré disk