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

,

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 .

[1] M. Berezowski, "Effect of Delay Time on the Generation of Chaos in Continuous Systems. One-Dimensional Model. Two-Dimensional Model—Tubular Chemical Reactor with Recycle," Chaos, Solitons and Fractals,12(1), 2001 pp. 83–89. doi:10.1016/S0960-0779(99)00171-X.