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Period modulations of the long secondary periods on the AGB stars

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.