TY - JOUR

T1 - The sunflower equation

T2 - 6th IFAC Conference on Analysis and Control of Chaotic Systems CHAOS 2021

AU - Smirnova, Vera B.

AU - Proskurnikov, Anton V.

AU - Zgoda, Iurii

N1 - Publisher Copyright: Copyright © 2021 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

PY - 2021

Y1 - 2021

N2 - In this paper, we consider a delayed counterpart of the mathematical pendulum model that is termed sunflower equation and originally was proposed to describe a helical motion (circumnutation) of the apex of the sunflower plant. The “culprits” of this motion are, on one hand, the gravity and, on the other hand, the hormonal processes within the plant, namely, the lateral transport of the growth hormone auxin. The first mathematical analysis of the sunflower equation was conducted in the seminal work by Somolinos (1978) who gave, in particular, a sufficient condition for the solutions' boundedness and for the existence of a periodic orbit. Although more than 40 years have passed since the publication of the work by Somolinos, the sunflower equation is still far from being thoroughly studied. It is known that a periodic solution may exist only for a sufficiently large delay, whereas for small delays the equation exhibits the same qualitative behavior as a conventional pendulum, and every solution converges to one of the equilibria. However, necessary and sufficient conditions for the stability of the sunflower equation (ensuring the convergence of all solutions) are still elusive. In this paper, we derive a novel condition for its stability, which is based on absolute stability theory of integro-differential pendulum-like systems developed in our previous work. As will be discussed, our estimate for the maximal delay, under which the stability can be guaranteed, improves the existing estimates and appears to be very tight for some values of the parameters.

AB - In this paper, we consider a delayed counterpart of the mathematical pendulum model that is termed sunflower equation and originally was proposed to describe a helical motion (circumnutation) of the apex of the sunflower plant. The “culprits” of this motion are, on one hand, the gravity and, on the other hand, the hormonal processes within the plant, namely, the lateral transport of the growth hormone auxin. The first mathematical analysis of the sunflower equation was conducted in the seminal work by Somolinos (1978) who gave, in particular, a sufficient condition for the solutions' boundedness and for the existence of a periodic orbit. Although more than 40 years have passed since the publication of the work by Somolinos, the sunflower equation is still far from being thoroughly studied. It is known that a periodic solution may exist only for a sufficiently large delay, whereas for small delays the equation exhibits the same qualitative behavior as a conventional pendulum, and every solution converges to one of the equilibria. However, necessary and sufficient conditions for the stability of the sunflower equation (ensuring the convergence of all solutions) are still elusive. In this paper, we derive a novel condition for its stability, which is based on absolute stability theory of integro-differential pendulum-like systems developed in our previous work. As will be discussed, our estimate for the maximal delay, under which the stability can be guaranteed, improves the existing estimates and appears to be very tight for some values of the parameters.

KW - Delays system

KW - Nonlinear system

KW - Stability

KW - Sunflower equation

UR - http://www.scopus.com/inward/record.url?scp=85120897032&partnerID=8YFLogxK

U2 - 10.1016/j.ifacol.2021.11.038

DO - 10.1016/j.ifacol.2021.11.038

M3 - Article

AN - SCOPUS:85120897032

VL - 54

SP - 135

EP - 140

JO - IFAC-PapersOnLine

JF - IFAC-PapersOnLine

SN - 2405-8971

IS - 17

Y2 - 27 September 2021 through 29 September 2021

ER -