An Algebraic Approach to the Classification of Centers in Polynomial Li´enard Systems

We consider the second-order Li´enard system Rx C f x‘ P x C gx‘ D 0; where f x‘ and gx‘ are polynomials, which we rewrite in the equivalent twodimensional form, P x D y; Py D −gx‘ − yf x‘: We show that the local question of whether a critical point of this system is a center can be expressed in terms of global conditions on f and g. Using these results, we give a simple classification of all such centers. We also address the problem of the coexistence of centers, foci, and limit cycles for these systems. Systems with degenerate centers are also considered.