With specific nonlinearity, discrete memristors can form excellent chaotic systems through different coupling models. This work proposes a step-wise coupling method for constructing a mapping model. Based on second-order coupling, a novel trigonometric step-wise discrete memristive (TSDM) map is constructed by coupling a sine discrete memristor with a cosine discrete memristor. The dynamical behaviors, tuned by parameters and initial values, are investigated using various numerical methods. It is found that there is a local offset behavior of the attractor in the TSDM map, which is highly sensitive to the initial value. The TSDM map was implemented using a microcontroller, and a TSDM map-based pseudo-random number generator was designed.
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