Periodic solutions of the sunflower equation: π‘₯̈+(π‘Ž/π‘Ÿ)π‘₯+(𝑏/π‘Ÿ)sinπ‘₯(𝑑-π‘Ÿ)=0

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Abstract
In 1967 Israelsson and Johnsson proposed a model for the geotropic circumnutations of Helianthus annus. The existence of a geotropic reaction time is reflected in the delay r r of the equation. Numerical computations suggested the existence of periodic solutions. In this paper, we prove the existence of periodic solutions for a range of the values of the parameters a , b , r a,b,r . We use Razumikhin-type functions to prove the boundedness of all solutions. We then prove the existence of periodic solutions of small amplitude using the Hopf bifurcation theorem. Finally, we use a fixed-point theorem on a cone to prove the existence of periodic solutions of large amplitude.