  30$\%$--50$\%$ of the luminous pulsating red-giant stars show light
variations of a longer period than the pulsation periods. Those periods are
called long secondary periods (LSP). There has been debated for many years but
the origin of the LSP is still unknown. To explain the LSP variations, there
have been many approaches in not only observations but also theoretical
studies. However, the invariance of the length of the LSPs has been
investigated little. Thus, we studied the temporal variations of the period by
performing the weighted wavelet-Z-transform analysis.
  Using the OGLE-III database, the $I$-band light curves of 6904 and 1945 LSP
candidates in the Large/Small Magellanic Clouds, respectively, were analyzed.
Most of our sample stars indicated that the period corresponding to the LSP was
constant during the observation term. However, 101 and 44 LSP stars in the LMC
and SMC, respectively, showed the signature of the temporal variations of the
LSPs. There were diversities of the period modulations i.e. monotonic increase
or decrease, or constant until the middle and then increase, etc. The
comet-like companion is one of the possible explanations for the LSP
variations, but this hypothesis cannot explain the period modulations because
the LSP is determined by the orbital period.
