Fractional Order Sunflower Equation: Stability, Bifurcation and Chaos

TL;DR

This paper introduces a fractional-order generalization of the sunflower equation, analyzing its stability, bifurcation, and chaos, revealing complex dynamics including multi-scroll chaotic attractors.

Contribution

It develops a fractional-order model of the sunflower equation and provides a detailed stability and bifurcation analysis, highlighting new dynamic behaviors such as chaos.

Findings

Identified stability regions and bifurcation points dependent on delay and fractional orders.

Discovered multi-scroll chaotic attractors under certain parameter conditions.

Established conditions for stability switches and critical delay values.

Abstract

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.