Abstract
In this paper we study the existence of limit cycle for cubic system (Ē)3, of Kolmogorov type with a conic algebraic trajectory
It has been proved in my former papers that (Ē)3 doesn't have any limit cycle on the whole plane lfb 2−ac≠0[4,5], Now we are investigating the case whereb 2−ac=0. We prove the sufficient and necessary formula (2) or (13) with which (Ē)3 must have a parabolic trajectoryF 2(x, y)=0. Then there 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 on the whole plane has no limit cycle.
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Huang, Q. Limit cycle for cubic system of Kolmogorov type with a conic algebraic trajectory. Acta Mathematicae Applicatae Sinica 5, 110–115 (1989). https://doi.org/10.1007/BF02009744
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DOI: https://doi.org/10.1007/BF02009744