  In this paper, we investigate the geometry of a general class of gradient
flows with multiple local maxima. we decompose the underlying space into
disjoint regions of attraction and establish the adjacency criterion. The
criterion states a necessary and sufficient condition for two regions of
attraction of stable equilibria to be adjacent. We then apply this criterion on
a specific type of gradient flow which has as many as n! local maxima. In
particular, we characterize the set of equilibria, compute the index of each
critical manifold and moreover, find all pairs of adjacent neighbors. As an
application of the adjacency criterion, we introduce a stochastic version of
the double bracket flow and set up a Markov model to approximate the sample
path behavior. The study of this specific prototype with its special structure
provides insight into many other difficult problems involving simulated
annealing.
