Maximum number of limit cycles of cubic Lienard differential system

Abstract

The number of limit cycles of the cubic Lie´nard polynomial differential system of the form x˙ = y, y˙ = - g(x)- f (x)y is examined, where f (x) is a polynomial of degree three and g(x), a polynomial of degree one and two. The accurate upper bound of the maximum number of limit cycles of this Lie´nard differential system is obtained. By using the first order averaging theory, this system is shown to bifurcate from the periodic orbits of the linear center x˙ = y, y˙= -x. The maximum number of limit cycles of the differential system is found to be unique.