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

Abstract

Various physical, ecological, economic, etc phenomena are governed by planar differential systems. Sub- sequently, several research studies are interested in the study of limit cycles because of their interest in the understanding of these systems. The aim of this paper is to investigate a class of quintic Kolmogorov systems, namely systems of the form x=xP4 (x;y); y= y Q4 (x; y) ; where P4 and Q4 are quartic polynomials. Within this class, our attention is restricted to study the limit cycle in the realistic quadrant {(x; y) 2 R2; x > 0; y > 0}. According to the hypothesises, the existence of algebraic or non-algebraic limit cycle is proved. Furthermore, this limit cycle is explicitly given in polar coordinates. Some examples are presented in order to illustrate the applicability of our result

A class of differential systems of even degree with exact non-algebraic limit cycles

“…In [6], an example of an explicit limit cycle which is not algebraic is given for n = 3. Bendjeddou in [3] provide a class of polynomial differential system of degree odd with explicit limit cycle non-algebraic.…”

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.…”

Two Non Algebraic Limit Cycles of a Class of Polynomial Differential Systems with Non-Elementary Equilibrium Point

“…Afterwards, A l-D o s a r y [1] gave a family of polynomial differential systems, with an explicit non-algebraic limit cycle, which generalizes the system studied in [9]. B e n d j e d d o u in [3] provided a class of polynomial differential system of degree odd with explicit non-algebraic limit cycle.…”