Summary: We have proposed an optimization method for a combinatorial optimization problem using replicator equations. To improve the solution further, a deterministic annealing algorithm may be applied. During the annealing process, bifurcations of equilibrium solutions will occur and affect the performance of the deterministic annealing algorithm. In this paper, the bifurcation structure of the proposed model is analyzed in detail. It is shown that only pitchfork bifurcations occur in the annealing process, and the solution obtained by the annealing is the branch uniquely connected with the uniform solution. It is also shown experimentally that in many cases, this solution corresponds to a good approximate solution of the optimization problem. Based on the results, a deterministic annealing algorithm is proposed and applied to the quadratic assignment problem to verify its performance.