  We address an iterative procedure that can be used to detect coarse-grained
hyperbolic unstable equilibria (saddle points) of microscopic simulators when
no equations at the macroscopic level are available. The scheme is based on the
concept of coarse timestepping [Kevrekidis et al., 2003] incorporating an
adaptive mechanism based on the chord method allowing the location of
coarse-grained saddle points directly. Ultimately, it can be used in a
consecutive manner to trace the coarse-grained open-loop saddle-node
bifurcation diagrams. We illustrate the procedure through two indicatively
examples including (i) a kinetic Monte Carlo simulation (kMC) of simple surface
catalytic reactions and (ii) a simple agent-based model, a financial caricature
which is used to simulate the dynamics of buying and selling of a large
population of interacting individuals in the presence of mimesis. Both models
exhibit coarse-grained regular turning points which give rise to branches of
saddle points.
