Maximum number of limit cycles of a general Lienard equation

We analyse the number of limit cycles of the general form of Lienard equation consisting of f(x), g(x), as polynomials of degree n and m respectively, and together with a small parameter ε. We determine an accurate upper bound of the maximum number of limit cycles that this Lienard equation can have. This is shown by bifurcating from the periodic orbits of the linear center ̇x = y, ̇y = −x, via the first order of the averaging theory.