Isochronous Centers of Lienard Type Equations and Applications

In this work we study the equation $(E) \ddot x + f(x) \dot x^2 + g(x) = 0$ 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 $(L_{D,F})$. Some classes of Kukles are also considered. Moreover, we classify a 5-parameters family of reversible cubic systems with isochronous centers. Key Words and phrases: period function, monotonicity, isochronicity, center, polynomial systems.