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.
H. Andersen and A. Johnsson, Entrainment of geotropic oscillations in hypocotyes of Helianthus annus, Physiol. Plant, 26, 44β61 (1972)
T. Baranetzki, Die kreisformige Nutation and das Winden der Stengel, Mem. Iβacad. imp. Sciences St. Petersbourg, ser. 7, 31, 1β73 (1883)
A. Casal and A. Somolinos, Estudio numerico de la ecuacion del Girasol, Revista de la Academia de Ciencias, Madrid, to appear
- Shui Nee Chow and Jack K. Hale, Periodic solutions of autonomous equations, J. Math. Anal. Appl. 66 (1978), no. 3, 495β506. MR 517743, DOI https://doi.org/10.1016/0022-247X%2878%2990250-0
M. Darwin, On the movements and habits of climbing plants, J. Linn. Soc. Bot. 9, 1β118 (1865)
H. Gradmann, Die Uberkrummungsbewegung der Ranken, Jn. Wiss. Bot. 60, 411β457 (1921)
- R. B. Grafton, A periodicity theorem for autonomous functional differential equations, J. Differential Equations 6 (1969), 87β109. MR 243176, DOI https://doi.org/10.1016/0022-0396%2869%2990119-3
- R. B. Grafton, Periodic solutions of certain LiΓ©nard equations with delay, J. Differential Equations 11 (1972), 519β527. MR 293207, DOI https://doi.org/10.1016/0022-0396%2872%2990064-2
- Jack K. Hale, Functional differential equations, Analytic theory of differential equations (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich., 1970) Springer, Berlin, 1971, pp. 9β22. Lecture Notes in Mat., Vol. 183. MR 0390425
- Jack Hale, Theory of functional differential equations, 2nd ed., Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences, Vol. 3. MR 0508721
D. Israelsson and A. Johnsson, A theory for circumnutations of Helianthus annus, Physiol. Plant 20, 957β976 (1967)
A. Johnsson, Geotropic responses in Helianthus and their dependence on the auxin ratio, Physiol. Plant 24, 419 (1971)
A. Johnsson and H.G. Karlsson, A feedback model for biological rhythms, J. Theor. Biol. 36. 153β201 (1972)
P. Lima, Hopf bifurcation for equations with infinite delays, Ph.D. Thesis, Brown University, 1977
M. Mohl, Uber den Bau und das Winden der Ranken und Schlingplanzen, Tubingen, 1827
- Roger D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl. (4) 101 (1974), 263β306. MR 361372, DOI https://doi.org/10.1007/BF02417109
- Roger D. Nussbaum, A global bifurcation theorem with applications to functional differential equations, J. Functional Analysis 19 (1975), no. 4, 319β338. MR 0385656, DOI https://doi.org/10.1016/0022-1236%2875%2990061-0
L . M. Palm, Uber das Winden der Planzen, Stuttgart, 1827
H. Andersen and A. Johnsson, Entrainment of geotropic oscillations in hypocotyes of Helianthus annus, Physiol. Plant, 26, 44β61 (1972)
T. Baranetzki, Die kreisformige Nutation and das Winden der Stengel, Mem. Iβacad. imp. Sciences St. Petersbourg, ser. 7, 31, 1β73 (1883)
A. Casal and A. Somolinos, Estudio numerico de la ecuacion del Girasol, Revista de la Academia de Ciencias, Madrid, to appear
S. Chow and J. K. Hale, Periodic solutions of autonomous equations, J. Math. Anal. Appl., to appear
M. Darwin, On the movements and habits of climbing plants, J. Linn. Soc. Bot. 9, 1β118 (1865)
H. Gradmann, Die Uberkrummungsbewegung der Ranken, Jn. Wiss. Bot. 60, 411β457 (1921)
R. Grafton, A periodicity theorem for autonomous functional equations, J. Diff. Eqs. 6, 87β109 (1969)
R. Grafton, Periodic solutions of certain Lienard equations with delay, J. Diff. Eqs. 11, 519β527 (1972)
J.K. Hale, Functional differential equations, Springer-Verlag, 1971
J.K. Hale, Theory of functional differential equations, Springer-Verlag, 1977
D. Israelsson and A. Johnsson, A theory for circumnutations of Helianthus annus, Physiol. Plant 20, 957β976 (1967)
A. Johnsson, Geotropic responses in Helianthus and their dependence on the auxin ratio, Physiol. Plant 24, 419 (1971)
A. Johnsson and H.G. Karlsson, A feedback model for biological rhythms, J. Theor. Biol. 36. 153β201 (1972)
P. Lima, Hopf bifurcation for equations with infinite delays, Ph.D. Thesis, Brown University, 1977
M. Mohl, Uber den Bau und das Winden der Ranken und Schlingplanzen, Tubingen, 1827
R. Nussbaum, Periodic solutions of some non-linear autonomous functional equations, Ann. Math. Pura Appl. 10, 263β306 (1974)
R. Nussbaum, A global bifurcation theorem with applications to functional differential equations, J. Functional Anal. 19 (1975)
L . M. Palm, Uber das Winden der Planzen, Stuttgart, 1827
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Article copyright:
© Copyright 1978
American Mathematical Society