An analysis is presented of the dynamic responses of nonlinear asynchronous iterations in multiprocessors in terms of the delays and the relaxation parameters. The actual distributed implementation is simulated, for analytical purposes, by means of systems of nonlinear difference equations. Local and global stability analyses of these systems are used to reveal the rich dynamical behavior of chaotic relaxations in the presence of nonlinearities and nonconvexities. We show the effect of the delays and relaxation parameters in the topology of the basins of attraction of different equilibrium points by constructing appropriate Lyapunov functions. The occurrence of bifurcation phenomena leading to chaotic behavior is shown. Finally, we study the phenomenon of the appearance of metastable equilibrium points, as functions of the communicational delays among the different processors.