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Recently, synthesis of gene regulatory networks having desired behavior has become of interest to many researchers and several studies have been done. There exist periodic phenomena in cells and those are considered to be generated by gene regulatory networks such as circadian rhythm. In this paper, we discuss a synthesis problem of gene regulatory networks possessing desired cyclic expression pattern sequences. It does not imply persistent oscillations of gene expression quantities that gene regulatory networks have cyclic expression pattern sequences. Therefore we have proposed a synthesis method in which periodicity of solution trajectories of gene regulatory networks is ensured by specifying passing points of the trajectories. However there does not always exist a gene regulatory network having a solution trajectory passing through the points. In this paper, we propose a synthesis method in which we introduce Poincarè maps for desired cyclic expression pattern sequences and make a Poincaré map have a fixed point. As a result, synthesized gene regulatory networks have desired cyclic expression pattern sequences together with their corresponding solution trajectories being periodic. Numerical experiments are carried out to illustrate the performance of the proposed method.