Oscillations of the sunflower equation

Abstract

Consider the delay differential equation \[ y ¨ ( t ) + α y ˙ ( t ) + β f ( y ( t − r ) ) = 0 , ( ∗ ) \ddot y\left ( t \right ) + \alpha \dot y\left ( t \right ) + \beta f\left ( {y\left ( {t - r} \right )} \right ) = 0, \qquad \left ( * \right ) \] where α , β \alpha , \beta , and r r are positive constants and f f is a continuous function such that \[ u f ( u ) > 0 f o r u ∈ [ − A , B ] , u ≠ 0 , a n d lim u → 0 f ( u ) u = 1 , uf\left ( u \right ) > 0 \qquad for u \in \left [ { - A, B} \right ], u \ne 0, and \lim \limits _{u \to 0} \frac {{f\left ( u \right )}}{u} = 1, \] where A A and B B are positive numbers. When f ( u ) = sin ⁡ u , ( ∗ ) 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.