  Metastable states in Ising spin-glass models are studied by finding iterative
solutions of mean-field equations for the local magnetizations. Two different
equations are studied: the TAP equations which are exact for the SK model, and
the simpler `naive-mean-field' (NMF) equations. The free-energy landscapes that
emerge are very different. For the TAP equations, the numerical studies confirm
the analytical results of Aspelmeier et al., which predict that TAP states
consist of close pairs of minima and index-one (one unstable direction) saddle
points, while for the NMF equations saddle points with large indices are found.
For TAP the barrier height between a minimum and its nearby saddle point scales
as (f-f_0)^{-1/3} where f is the free energy per spin of the solution and f_0
is the equilibrium free energy per spin. This means that for `pure states', for
which f-f_0 is of order 1/N, the barriers scale as N^{1/3}, but between states
for which f-f_0 is of order one the barriers are finite and also small so such
metastable states will be of limited physical significance. For the NMF
equations there are saddles of index K and we can demonstrate that their
complexity Sigma_K scales as a function of K/N. We have also employed an
iterative scheme with a free parameter that can be adjusted to bring the system
of equations close to the `edge of chaos'. Both for the TAP and NME equations
it is possible with this approach to find metastable states whose free energy
per spin is close to f_0. As N increases, it becomes harder and harder to find
solutions near the edge of chaos, but nevertheless the results which can be
obtained are competitive with those achieved by more time-consuming computing
methods and suggest that this method may be of general utility.
