This paper proposes a method to linguistically summarise the local periodic components of a time series: it identifies subparts of the data which are periodic, together with their periodicity degree and period, and provides a linguistic description thereof. The generated sentences can be illustrated by the example " Approximately from March to June, the series is highly periodic with a period of exactly 2 weeks ". The method proposed to identify local periodic zones relies on the determination of relevant auto-adaptive windows, based on an analytical expression of the probability distribution of the considered periodicity criterion. The linguistic description generation, in the protoform approach framework, expresses three core aspects of the identified periodic intervals, namely their time context or localisation in time, their periodicity and their period. Intensive experiments performed on both artificial and real data validate the proposed method.