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