Periodic solutions of the sunflower equation: 𝑥̈+(𝑎/𝑟)𝑥+(𝑏/𝑟)sin𝑥(𝑡-𝑟)=0

Somolinos, Alfredo S.

doi:10.1090/qam/465265

Periodic solutions of the sunflower equation: $\ddot x+(a/r) x+(b/r)\sin x(t-r)=0$

Author: Alfredo S. Somolinos
Journal: Quart. Appl. Math. 35 (1978), 465-478
MSC: Primary 92A05; Secondary 34C25, 58F10
DOI: https://doi.org/10.1090/qam/465265
MathSciNet review: 465265
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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 $r$ 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 $a,b,r$. 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.

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