Exploring delay dynamics with a programmable electronic delay circuit

Delay dynamics occur in a wide variety of natural and man-made systems. Even simple delay systems can generate complex dynamics whose exploration is rewarding. To allow such exploration as part of advanced undergraduate laboratory courses and be able to utilize systems that operate at convenient timescales, it is necessary to delay analog signals by several milliseconds. In this paper, we describe an implementation of a programmable digital circuit capable of delaying DC-coupled analog signals up to 262 ms at a 1 MHz sampling rate. The initial history of the system may also be arbitrarily programmed, enabling the study of transient behavior. As an application, we discuss the use of this programmable delay in a feedback circuit that produces period-four triangular solutions, in complete agreement with theoretical predictions.