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