Oscillations of the sunflower equation

Consider the delay differential equation \[ \ddot y\left ( t \right ) + \alpha \dot y\left ( t \right ) + \beta f\left ( {y\left ( {t - r} \right )} \right ) = 0, \qquad \left ( * \right )\] where $\alpha , \beta$, and $r$ are positive constants and $f$ is a continuous function such that \[ uf\left ( u \right ) > 0 \qquad for u \in \left [ { - A, B} \right ], u \ne 0, and \lim \limits _{u \to 0} \frac {{f\left ( u \right )}}{u} = 1,\] where $A$ and $B$ are positive numbers. When $f\left ( u \right ) = \sin u, \left ( * \right )$ is the so-called “sunflower” equation, which describes the motion of the tip of the sunflower plant.