Nonlinearity is an underlying property of process systems that allows for time-average performance enhancement with periodic forcing of external parameters. One can use the well-known pi-criterion to obtain a sufficient condition for optimal periodic operation, but it is valid only for weakly nonlinear systems with infinitesimal forcing amplitudes. Among such process systems, bioreactors often exhibit highly nonlinear dynamics, thus posing difficulties for a systematic analysis of their periodic operation. It becomes desirable to understand how inherent nonlinearities would contribute to performance improvement if periodic forcing is applied to such processes. This paper explores how nonlinear characteristics affect periodic process operation in the context of bioreactor systems. By taking advantage of iterated integrals and shuffle algebra, an analytical solution of a nonlinear bioprocess is approximated with the functional expansion method. The resulting Laplace-Borel transform of the nonlinear system facilitates the expression of the solution in the frequency domain based on the nonlinear transfer function. The method enables the separation of the transient dynamics and the stationary periodic behavior, facilitating the analysis of periodic operations and leading to a generalizable platform. Specifically, the optimal periodic operation problem is solved by finding the proper forcing amplitude and frequency to maximize the offset. Compared with the pi-criterion, our method is proven to be globally valid for any forcing frequency and amplitude, so long as the process is globally stable. (C) 2017 Elsevier Ltd. All rights reserved.