Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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.


References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/qam/465265
Article copyright: © Copyright 1978 American Mathematical Society

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