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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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] Shui Nee Chow and Jack K. Hale, Periodic solutions of autonomous equations, J. Math. Anal. Appl. 66 (1978), no. 3, 495–506. MR 517743, https://doi.org/10.1016/0022-247X(78)90250-0
  • [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. B. Grafton, A periodicity theorem for autonomous functional differential equations., J. Differential Equations 6 (1969), 87–109. MR 0243176, https://doi.org/10.1016/0022-0396(69)90119-3
  • [8] R. B. Grafton, Periodic solutions of certain Liénard equations with delay, J. Differential Equations 11 (1972), 519–527. MR 0293207, https://doi.org/10.1016/0022-0396(72)90064-2
  • [9] Jack K. Hale, Functional differential equations, Analytic theory of differential equations (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich., 1970) Springer, Berlin, 1971, pp. 9–22. Lecture Notes in Mat., Vol. 183. MR 0390425
  • [10] Jack Hale, Theory of functional differential equations, 2nd ed., Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences, Vol. 3. 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] Roger D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl. (4) 101 (1974), 263–306. MR 0361372, https://doi.org/10.1007/BF02417109
  • [17] Roger D. Nussbaum, A global bifurcation theorem with applications to functional differential equations, J. Functional Analysis 19 (1975), no. 4, 319–338. MR 0385656
  • [18] L . M. Palm, Uber das Winden der Planzen, Stuttgart, 1827

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 92A05, 34C25, 58F10

Retrieve articles in all journals with MSC: 92A05, 34C25, 58F10

Additional Information

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

American Mathematical Society