On the Dynamics of a Class of Kolmogorov Systems

On the non existence of periodic orbits for a class of two dimensional Kolmogorov systems

“…where f 1 (θ) = 1+ 3 4 sin 2θ − 1 8 sin 4θ, f 2 (θ) = (3 − cos θ sin θ) ln 9 2 + 1 2 cos 2θ and f 3 (θ) = cos 2 θ sin 2 θ. In the positive quadrant (x > 0, y > 0) it is the case (a) of the Theorem 1, then the Kolmogorov system (8) has the first integral…”

“…In the qualitative theory of planar dynamic systems [4,7,8,9,16], one of the most important topics is related to the second part of the unsolved Hilbert 16th problem [13]. There is a huge literature about limit cycles, most of them deal essentially with their detection, their number and their stability and rare are papers concerned by giving them explicitly [1,2,3,11,21]. System (1) is integrable on an open set Ω of R 2 if there exists a non constant C 1 function H : Ω → R, called a first integral of the system on Ω , which is constant on the trajectories of the system (1) contained in Ω, i.e., if…”

Explicit Form for the First Integral and Limit Cycles of a Class of Planar Kolmogorov Systems

“…There is a huge literature about limit cycles, and most of them essentially deal with their detection, the number and the stability. However, ther are a very few papers concerning explicitly forms of limit cycles [16][17][18][19].…”