  Maxmin-$\omega$ dynamical systems were previously introduced as a
generalization of dynamical systems expressed by tropical linear algebra. To
describe steady states of such systems one has to study an eigenproblem of the
form $A\otimes_{\omega} x=\lambda+x$ where $\otimes_{\omega}$ is the
maxmin-$\omega$ matrix-vector multiplication. This eigenproblem can be viewed
in more general framework of nonlinear Perron-Frobenius theory. However,
instead of studying such eigenspaces directly we develop a different approach:
over-approximation by zones. These are traditionally convex sets of special
kind which proved to be highly useful in computer science and also relevant in
tropical convexity. We first construct a sequence of zones over-approximating a
maxmin-$\omega$ eigenspace. Next, the limit of this sequence is refined in a
heuristic procedure, which yields a refined zone and also the eigenvalue
$\lambda$ with a high success rate. Based on the numerical experiments, in
successful cases there is a column of the difference bound matrix (DBM)
representation of the refined zone which yields an eigenvector.
