Periodic trend and fluctuations: The case of strong correlation

We study the effects of an external periodic perturbation on a Poisson rate process, with special attention to the perturbation-induced sojourn-time patterns. We show that these patterns correspond to turning a memory-less sequence into a sequence with memory. The memory effects are stronger the slower the perturbation. The adoption of a de-trending technique, applied with no caution, might generate the impression that no fluctuation–periodicity correlation exists. We find that this is due to the fact that the perturbation-induced memory is a global property and that the result of a local in time analysis would not find any memory effect, insofar as the process under study is locally a Poisson process. We find that an efficient way to detect this memory effect is to analyze the moduli of the de-trended sequence. We turn the sequence to analyze into a diffusion process, and we evaluate the Shannon entropy of the resulting diffusion process. We find that both the original sequence and the suitably processed de-trended sequence yield the same dependence of entropy on time, namely, an initial scaling larger than ordinary scaling, and a sequel of weak oscillations, which are a clear signature of the external perturbation, in both cases. This is a clear indication of the fluctuation–periodicity correlation.