  The theory of slow-fast gradient systems leads in a natural way to
non-equilibrium steady states, because on the slow time scale the fast
subsystem stays in steady states that are controlled by the interaction with
the slow system. Using the theory of convergence of gradient systems depending
on a small paramter $\varepsilon$ (here the ratio between the slow and the fast
time scale) in the sense of the "energy-dissipation principle" (EDP) shows that
there is a natural characterization of these non-equilibrium steady states as
saddle points of a Lagrangian where the slow variables are fixed. We give
applications to slow-fast reaction-diffusion systems based on the so-called
cosh-type gradient structure for reactions. It is shown that two binary
reaction give rise to a ternary reaction with a state-dependent reaction
coefficient. Moreover, we show that a reaction-diffusion equation with a thin
membrane-like layer convergences to a transmission condition, where the
formerly quadratic dissipation potential for diffusion convergences to a
cosh-type dissipation potential for the transmission in the membrane limit.
