Resumen de Studying of Behavior at Infinity of Vector Fields on Poincaré’s Sphere: Revisited

Abdulla Azamov, Shakhzod Suvanov, Asliddin Tilavov

• Common used Poincaré’s method to study of qualitative behavior of polynomial systems at infinity is revisited. An effect called “loss of projectivity” in dynamical systems of even degree is accented. It is given an interpretation of Lefschetz’s equation in terms of Pfaff forms and a modified Lefschetz’s equation is suggested which allows to get projective extension for dynamical systems of both odd and even degrees. The constructions imply that every polynomial vector field in Rd can be extended onto projective space PRd (d = 2, 3, 4, ...) saving polynomiality.