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