Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve
- 48 Downloads
In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x2 + y2 + 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincaré disc.
KeywordsQuadratic system complex ellipse invariant algebraic curves phase portrait Poincaré disc
MR(2010) Subject Classification34C05
Unable to display preview. Download preview PDF.
- Kapteyn, W.: On the midpoints of integral curves of differential equations of the first degree (Dutch). Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, 19, 1446–1457 (1911)Google Scholar
- Kapteyn, W.: New investigations on the midpoints of integrals of differential equations of the first degree (Dutch). Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, 20, 1354–1365 (1912), 21, 27–33 (1013)Google Scholar
- Korol, N. A.: The integral curves of a certain differential equation (in Russian). Minsk. Gos. Ped. Inst. Minsk, 47–51 (1973)Google Scholar
- Qin, Y. X.: On the algebraic limit cycles of second degree of the differential equation dy/dx = Σ0≤i+j≤2 a ij x i y j/Σ0≤i+j≤2 b ij x i y j. Acta Math. Sin., 8, 23–35 (1958)Google Scholar
- Vdovina, E. V.: Classification of singular points of the equation y’ = (a 0 x 2 +a 1 xy +a 2 y 2)/(b 0 x 2 +bnxy + b 2 y 2) by Forster’s method (in Russian). Differential Equations, 20, 1809–1813 (1984)Google Scholar