We are interested in the asymptotic
behaviors of a discrete-time neural network. This network admits
transiently chaotic behaviors which provide global searching
ability in solving combinatorial optimization problems. As the
system evolves, the variables corresponding to temperature in the
annealing process decrease, and the chaotic behaviors vanish. We
shall find sufficient conditions under which evolutions for the
system converge to a fixed point of the system. Attracting sets
and uniqueness of fixed point for the system are also addressed.
Moreover, we extend the theory to the neural networks with
cycle-symmetric coupling weights and other output functions. An
application of this annealing process in solving travelling
salesman problems is illustrated.