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?)

  • [1] H. Andersen and A. Johnsson, Entrainment of geotropic oscillations in hypocotyes of Helianthus annus, Physiol. Plant, 26, 44-61 (1972)
  • [2] T. Baranetzki, Die kreisformige Nutation and das Winden der Stengel, Mem. I'acad. imp. Sciences St. Petersbourg, ser. 7, 31, 1-73 (1883)
  • [3] A. Casal and A. Somolinos, Estudio numerico de la ecuacion del Girasol, Revista de la Academia de Ciencias, Madrid, to appear
  • [4] S. Chow and J. K. Hale, Periodic solutions of autonomous equations, J. Math. Anal. Appl., to appear MR 517743
  • [5] M. Darwin, On the movements and habits of climbing plants, J. Linn. Soc. Bot. 9, 1-118 (1865)
  • [6] H. Gradmann, Die Uberkrummungsbewegung der Ranken, Jn. Wiss. Bot. 60, 411-457 (1921)
  • [7] R. Grafton, A periodicity theorem for autonomous functional equations, J. Diff. Eqs. 6, 87-109 (1969) MR 0243176
  • [8] R. Grafton, Periodic solutions of certain Lienard equations with delay, J. Diff. Eqs. 11, 519-527 (1972) MR 0293207
  • [9] J.K. Hale, Functional differential equations, Springer-Verlag, 1971 MR 0390425
  • [10] J.K. Hale, Theory of functional differential equations, Springer-Verlag, 1977 MR 0508721
  • [11] D. Israelsson and A. Johnsson, A theory for circumnutations of Helianthus annus, Physiol. Plant 20, 957-976 (1967)
  • [12] A. Johnsson, Geotropic responses in Helianthus and their dependence on the auxin ratio, Physiol. Plant 24, 419 (1971)
  • [13] A. Johnsson and H.G. Karlsson, A feedback model for biological rhythms, J. Theor. Biol. 36. 153-201 (1972)
  • [14] P. Lima, Hopf bifurcation for equations with infinite delays, Ph.D. Thesis, Brown University, 1977
  • [15] M. Mohl, Uber den Bau und das Winden der Ranken und Schlingplanzen, Tubingen, 1827
  • [16] R. Nussbaum, Periodic solutions of some non-linear autonomous functional equations, Ann. Math. Pura Appl. 10, 263-306 (1974) MR 0361372
  • [17] R. Nussbaum, A global bifurcation theorem with applications to functional differential equations, J. Functional Anal. 19 (1975) MR 0385656
  • [18] L . M. Palm, Uber das Winden der Planzen, Stuttgart, 1827

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

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