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

arXiv Mathematics eprints
 Pub Date:
 March 2001
 arXiv:
 arXiv:math/0103180
 Bibcode:
 2001math......3180R
 Keywords:

 Dynamical Systems;
 Classical Analysis and ODEs;
 34C25;
 34C35
 EPrint:
 25 pages