  Searches for periodicity in time series are often done with models of
periodic signals, whose statistical significance is assessed via false alarm
probabilities or Bayes factors. However, a statistically significant periodic
model might not originate from a strictly periodic source. In astronomy in
particular, one expects transient signals that show periodicity for a certain
amount of time before vanishing. This situation is encountered for instance in
the search for planets in radial velocity data. While planetary signals are
expected to have a stable phase, amplitude and frequency - except when strong
planet-planet interactions are present - signals induced by stellar activity
will typically not exhibit the same stability. In the present article, we
explore the use of periodic functions multiplied by time windows to diagnose
whether an apparently periodic signal is truly so. We suggest diagnostics to
check whether a signal is consistently present in the time series, and has a
stable phase, amplitude and period. The tests are expressed both in a
periodogram and Bayesian framework. Our methods are applied to the Solar
HARPS-N data as well as HD 215152, HD 69830 and HD 13808. We find that (i) the
HARPS-N Solar data exhibits signals at the Solar rotation period and its first
harmonic ($\sim$ 13.4 days). The frequency and phase of the 13.4 days signal
appear constant within the estimation uncertainties, but its amplitude presents
significant variations which can be mapped to activity levels. (ii) as
previously reported, we find four, three and two planets orbiting HD 215152, HD
69830 and HD 13808.
