Singular Cycles of a Class of Quadratic Systems with a Third Order Fine Focus

摘要

The quadratic system dx/dt = -y + lx squared + 5axy, dy/dt = x + ax squared +3lxy is studied. It is proved that when the origin is not a center, a singular cycle passing through a saddle and surrounding the origin does not exist nor does there exist a singular cycle with an infinite element surrounding the origin. Complete analysis of the system shows that there does not exist a limit cycle in the whole plane system. The nonexistence of singular cycles in the system and its phase portrait are studied.