On the classification of Liénard systems with amplitude-independent periods

Abstract We consider conditions under which the second-order differential equation x +f(x) x +g(x)=0 has a family of periodic orbits with constant period. This condition is equivalent to seeking conditions under which the two-dimensional autonomous system x =y, y =−g(x)−f(x)y has a center with constant period: i.e., an isochronous center. In turn, this is equivalent to the latter system being locally linearizable. We give a simple necessary and sufficient condition for this when f and g are analytic functions. In the case where when f and g are polynomials, we show that this reduces to a finitely determinable system of equations. A complete classification is given of all such systems of degree 34 or less.