This paper introduces discrete Euler processes and shows their application in detecting and forecasting cycles in non-stationary data where periodic behavior changes approximately linearly in time. A discrete Euler process becomes a classical stationary process if 'time' is transformed properly. By moving from one time domain to another, one may deform certain time-varying data to non-time-varying data. With these non-time-varying data on the deformed timescale, one may use traditional tools to do parameter estimation and forecasts. The obtained results then can be transformed back to the original timescale. For datasets with an underlying discrete Euler process, the sample M-spectrum and the spectra estimator of a Euler model (i.e., EAR spectral) are used to detect cycles of a Euler process. Beam response and whale data are used to demonstrate the usefulness of a Euler model. Copyright © 2008 John Wiley & Sons, Ltd.