Effect of the time delay on the stability and instability of the logistic map

TL;DR

This paper investigates how time delay affects the stability of the logistic map, providing mathematical proofs for conjectures about when delays lead to stability or instability.

Contribution

It formulates and proves two conjectures on the impact of time delay on the stability and instability of the logistic map.

Findings

Delay can either stabilize or destabilize the logistic map.

Mathematical proofs confirm the conditions for stability and instability.

Results clarify the ambiguous role of delay in logistic models.

Abstract

A proper discretization of the logistic differential equation, which is preserving these two distinct equilibrium solutions and their unstability and stability, suggest that we need to examine the time delay of the logistic map. According to Murray, the effect of delay in models is "usually" to increase the potential for instability. However the word "usually" is really ambiguous. In this paper, we mathematically formulate and prove the two conjectures about stability and instability.