Isochronous centers of Lienard type equations and applications

In this work we study Eq. (E) ¨x +f (x) ˙x2 +g(x) = 0 with a center at 0 and investigate conditions of its isochronicity. When f and g are analytic (not necessary odd) a necessary and sufficient condition for the isochronicity of 0 is given. This approach allows us to present an algorithm for obtained conditions for a point of (E) to be an isochronous center. In particular, we find again by another way the isochrones of the quadratic Loud systems (LD,F ). We also classify a 5-parameters family of reversible cubic systems with isochronous centers. © 2006 Elsevier Inc. All rights reserved