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

We consider the second-order equation where f and g are polynomials with deg f,deg g 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 Liénard system, Assuming the origin is not a centre, we show that if either f or g are quadratic, then this number is . If f or g are cubic we show that this number is , for all 1n50. The results also hold for generalized Liénard systems.