Monotonicity of the period function for some planar differential systems. Part II: Liénard and related systems

A. Raouf Chouikha¹

¹ LAGA Université Paris 13 93430 Villetaneuse, France

Applicationes Mathematicae, Tome 32 (2005) no. 4, pp. 405-424

Résumé

We are interested in conditions under which the two-dimensional autonomous system $$ \dot x=y,\ \quad \dot y =-g(x) - f(x)y, $$ has a local center with monotonic period function. When $f$ and $g$ are (non-odd) analytic functions, Christopher and Devlin [C-D] gave a simple necessary and sufficient condition for the period to be constant. We propose a simple proof of their result. Moreover, in the case when $f$ and $g$ are of class $C^3$, the Liénard systems can have a monotonic period function in a neighborhood of $0$ under certain conditions. Necessary conditions are also given. Furthermore, Raleigh systems having a monotonic (or non-monotonic) period are considered.

Détail

DOI : 10.4064/am32-4-4

Keywords: interested conditions under which two dimensional autonomous system dot quad dot g has local center monotonic period function non odd analytic functions christopher devlin c d gave simple necessary sufficient condition period constant propose simple proof their result moreover class nard systems have monotonic period function neighborhood under certain conditions necessary conditions given furthermore raleigh systems having monotonic non monotonic period considered