Scalar Delay Logistic Equation

time t

50.

time delay τ

1

parameter a

4

parameter σ

0.2

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

τ

is the time delay and

a

and

σ

are positive constants. The values of the parameters generating bifurcations can be determined analytically [1]. Assuming the parameter values

a=3.9

,

τ=1

, chaos occurs for

σ<0.29

approximately; for large values of

σ

all oscillations disappear. Also interesting is the effect of time delay on the generation of chaos: when

a=3.9

and

σ=0.2

, all oscillations disappear at low values of

τ

. A necessary and sufficient condition for generating oscillations is

a>1+

1+

2

(σω)

, where the frequency

ω

is determined from the relationship

τ+arctan(σ,ω)=π

.