Kolmogorov System with Explicit Hyperbolic Limit Cycle

Abstract

A class of Kolmogorov differential system is introduced.We show that under suitable assumptions on parameters, an algebraic hyprbolic limit cycle can occur, the explicit expression of this limit cycle is given.

A survey on algebraic and explicit non-algebraic limit cycles in planar differential systems

“…Later on many other papers have been published providing explicit non-algebraic limit cycles for several polynomial differential systems, but in all these papers the explicit non-algebraic limit cycles are for polynomial differential systems of degree larger than or equal to 3, see [6,7,8,9,10,11,12,13,15,16], the authors of these papers are Aziza, Bendjeddou, Benterki, Benyoucef, Berbache, Boukoucha, Boukoucha, Cheurfa, Grazem and Salhi.…”

A Class of Quintic Kolmogorov Systems with Explicit Non-algebraic Limit Cycle

“…For instance, Bendjeddou, Cheurfa and Berbache in [2] showed that the quartic system admits the circle as an invariant curve which corresponds of course to the limit cycle. In the same context, Benyoucef and Bendjeddou studied in [7,8] two polynomial systems of any degrees. They showed in the first paper that the considered system can admit up to four algebraic limit cycles in the plane and in the second one the system can admit a unique algebraic limit cycle in the first quadrant.…”