It is shown analytically that pulse solitary waves in a chain with Lennard-Jones type nearest neighbor interaction are strongly localized and marginally stable in the high energy limit. In a damped and periodically driven chain we obtain numerically families of states whose behavior is similar to that of equally many oscillators. We observe a period doubling sequence in a one-solitary wave family and bifurcation to (quasi-) periodic motion in a family of two solitary waves. We conclude that the damped and driven chain admits asymptotically stable states living on a low-dimensional manifold in phase space. These results depend sensitively on the shape of the driving term.