Small-amplitude limit cycle bifurcations for Lienard systems with quadratic or cubic damping or restoring forces

We consider the second-order equation x + f(x)(x)over dot + g(x) = 0, (g(0) = 0, g'(0) > 0), where f and g are polynomials with deg f, deg g less than or equal to n. Our interest is in the maximum number of isolated periodic solutions which can bifurcate from the steady state solution x = 0. Alternatively, this is equivalent to seeking the maximum number of limit cycles which can bifurcate from the origin for the Lienard system, (x)over dot = y, (y)over dot = -g(x) - yf(x). Assuming the origin is not a centre, we show that if either f or g are quadratic, then this number is [2n+1/3]. If f or g are cubic we show that this number is 2[3(n+2)/8], for all 1 < n less than or equal to 50. The results also hold for generalized Lienard systems.