摘要
In this paper we study the existence of limit cycle for cubic system (E)_3, of Kolmogorov typewith a conic algebraic trajectoryF_2(x,y)=ax^2+2bxy+cy^2+dx+ey+f=0 It has been proved in my former papers that (E)_3 doesn't have any limit cycle on the whole planeIf b^2-ac≠0, Now we are investigating the case where b^2-ac=0. We prove the sufficient andnecessary formula (2) or (13) witb which (E)_3 must have a parabolic trajectory F_2(x,y)=0. Thenthere will not be any limit cycle on the full plane. On the basis of this, we conclude: The cubic system of Kolmogorov type with a non-degenerated quadratic algebraic trajectory onthe whole plane has no limit cycle.
In this paper we study the existence of limit cycle for cubic system (E)_3, of Kolmogorov typewith a conic algebraic trajectoryF_2(x,y)=ax^2+2bxy+cy^2+dx+ey+f=0 It has been proved in my former papers that (E)_3 doesn't have any limit cycle on the whole planeIf b^2-ac≠0, Now we are investigating the case where b^2-ac=0. We prove the sufficient andnecessary formula (2) or (13) witb which (E)_3 must have a parabolic trajectory F_2(x,y)=0. Thenthere will not be any limit cycle on the full plane. On the basis of this, we conclude: The cubic system of Kolmogorov type with a non-degenerated quadratic algebraic trajectory onthe whole plane has no limit cycle.
