# Scalar Delay Logistic Equation

Scalar Delay Logistic Equation

This Demonstration shows the solution of a simple scalar logistic delay equation that has found application in chemical engineering problems:

σx'(t)+x(t)=ax(t-τ)(1-x(t-τ))

where is the time delay and and are positive constants. The values of the parameters generating bifurcations can be determined analytically [1]. Assuming the parameter values , , chaos occurs for approximately; for large values of all oscillations disappear. Also interesting is the effect of time delay on the generation of chaos: when and , all oscillations disappear at low values of . A necessary and sufficient condition for generating oscillations is , where the frequency is determined from the relationship .

τ

a

σ

a=3.9

τ=1

σ<0.29

σ

a=3.9

σ=0.2

τ

a>1+

1+

2

(σω)

ω

τ+arctan(σ,ω)=π