The Period Function of Second Order Differential Equations

Abstract

We interest in the behaviour of the period function for equations of the type $u^{\prime \prime }+g(u)=0$ and $u^{\prime \prime }+f(u)u^{\prime }+g(u)=0$ with a center at the origin 0. $g$ is a function of class $C^k$. For the conservative case, if $k\ge 2$ one shows that the Opial criterion is the better one among those for which these the necessary condition $g^{\prime \prime }(0)=0$ holds. In the case where $f$ is of class $C^1$ and $k\ge 3$, the Lienard equations $u^{\prime \prime }+f(u)u^{\prime }+g(u)=0$ may have a monotonic period function if $g^{\prime }(0)g^{(3)}(0)-5/3{g^{\prime \prime }}^2(0)-2/3{f^{\prime }}^2(0)g^{\prime }(0)\ne 0$ in a neighborhood of 0. {\it Key Words and phrases:} period function, monotonicity, isochronicity, Lienard equation, polynomial systems.

25 pages