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There exist periodic phenomena in cells and the phenomena are considered to be generated by gene regulatory networks. Therefore synthesis of gene regulatory networks having desired periodic behavior has become of interest to many researchers. In this paper, we discuss a synthesis problem of gene regulatory networks having desired periodic behavior. The desired behavior are given by cyclic expression pattern sequences. We proposed a synthesis method for realizing a periodic solution trajectory exhibiting a desired cyclic expression pattern sequence by using Poincarè map. In the method, we choose a point and make it become a fixed point of a Poincaré map. However it is not always possible to make the chosen point become a fixed point. In this paper we propose a synthesis method without choosing a fixed point. We evaluate the method by performing numerical experiments.