The sunflower equation: novel stability criteria

Abstract

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.