On the exact limit cycle for some class of planar differential systems

Abstract

Inspired by a recent paper of Giacomini et al [3], we give the exact expression of the limit cycles for a class of two-dimensional differential sytems. We study also the uniqueness of such limit cycles. An application to Liénard equation and several examples are given at the end.2000 Mathematics Subject Classification: 34C25.

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

“…A limit cycle of system (1.1) is an isolated periodic solution in the set of all periodic solutions of system (1.1). If a limit cycle is contained in the zero level set of a polynomial function, see for example, [ [1], [4], [5], [9], [11]], then we say that it is algebraic, otherwise it is called non-algebraic see for example ([2], [4], [8], [10]). The topic of limit cycles is interesting both in mathematics and in science and many models from physics, engineering, chemistry, biology, economics,..., were displayed as differential systems with limit cycles.…”

Explicit limit cycles of a class of Kolmogorov system

“…This paper is a contribution in this direction, to determine the number of limit cycles and to give their explicit form. Motivated by the recent publication of some research papers exhibiting planar polynomial systems with one or more algebraic limit cycles analytically given (see for instance A. Bendjeddou and R. Cheurfa [1], [2], S.Benyoucef, A, Barbach and A.Bendjeddou, [3], S. Benyoucef…”

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

“…In the literature, we can find also another interesting but even more difficult problem is to give an explicit expression of a limit cycle. The limit cycles previously known in an explicit way were algebraic see [3,4,15].…”