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We consider a class of bilinear weakly contractive systems that entrain toperiodic excitations. Entrainment is important in many natural and artificialprocesses. For example, in order to function properly synchronous generatorsmust entrain to the frequency of the electrical grid, and biological organismsmust entrain to the 24h solar day. A dynamical system has a positive gain ofentrainment (GOE) if entrainment also yields a larger output, on average. Thisproperty is important in many applications from the periodic operation ofbioreactors to the periodic production of proteins during the cell cycledivision process. We derive a closed-form formula for the GOE to first-order inthe control perturbation. This is used to show that in the class of systemsthat we consider the GOE is always a higher-order phenomenon. We demonstratethe theoretical results using two applications: the master equation and a modelfrom systems biology called the ribosome flow model, both with time-varying andperiodic transition rates.

On the gain of entrainment in a class of weakly contractive bilinear  control systems with applications to the master equation and the ribosome  flow model