  The sunflower equation describes the motion of the tip of a plant due to the
auxin transportation under the influence of gravity. This work proposes the
fractional-order generalization to this delay differential equation. The
equation contains two fractional orders and infinitely many equilibrium points.
The problem is important because the coefficients in the linearized equation
near the equilibrium points are delay-dependent. We provide a detailed
stability analysis of each equilibrium point using linearized stability. We
find the boundary of the stable region by setting the purely imaginary value to
the characteristic root. This gives the conditions for the existence of the
critical values of the delay at which the stability properties change. We
observed the following bifurcation phenomena: stable for all the delay values,
a single stable region in the delayed interval, and a stability switch. We also
observed a multi-scroll chaotic attractor for some values of the parameters.
