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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
.
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