Existence and construction of some time‐periodic solutions of a class of reaction‐diffusion equations is described. These give examples of center‐structures from which emanate trains of waves with outward directed group velocity, in an infinite one‐dimensional spatial domain. A discrete set of distinct types of center‐structures is found. Similar results are found for sufficiently large finite regions with impermeable boundaries. The existence of many other time periodic solutions corresponding to spatially infinite irregular arrays of center‐structures of different types is also demonstrated for these systems. Some numerical examples ar