Searching for cycle and time-series I receive 339 results. Cycles are an important pattern in time-series analysis and therefore I think that they should receive their proper tag.
Edit: The difference between a cycle and a seasonality is that a seasonality always appears after the same time period. One example is that Christmas always appears once a year in December. Also every Sunday Christians celebrate Sunday. However a cycle appears in irregular time periods, e.g., business cycles where recession appears more or less twice a decade are an example.
See also here:
A seasonal pattern exists when a series is influenced by seasonal factors (e.g., the quarter of the year, the month, or day of the week). Seasonality is always of a fixed and known period.
A cyclic pattern exists when data exhibit rises and falls that are not of fixed period. The duration of these fluctuations is usually of at least 2 years.
@robjhyndman what do you think as an expert in this area?