On the limit cycles of some classes of kolmogorov differential systems

Abstract

Di erential equations were invented by Newton (1642-1727). It is the beginning of modern physics and the use of analysis to solve the law of universal gravitation leading to the ellipsity of the orbits of the planets in the solar system. Leibniz (1646-1716) erects analysis in autonomous discipline but it is necessary to await the work of Euler (1707-1783) and Lagrange (1736-1813) to see appear the methods allowing the resolution of the linear equations. The signi cant development of the theory of dynamical systems during this century has helped to develop methods of studying the properties of their solutions.The direct solution to the di erential system is usually di cult or impossible. Since Andronov (1932), three di erent approaches have traditionally been used to study dynamic systems: qualitative methods, analytical methods and numerical methods the most important of which are qualitative methods. One of the main problems in the qualitative theory of di erential equations is the study of the integrability (the rst integrals) and the limit cycles of polynomial planar di erential systems. The notion of rst integral appeared for the rst time in the work of G. Dar- boux (1842-1917) [7] in 1878. He built so-called general integrals for ordinary rst order di erential equations, having on many invariant algebraic curves.