The Fermi-Pasta-Ulam \ensuremath{\alpha} model of harmonic oscillators with cubic anharmonic interactions is studied from a statistical mechanical point of view. Systems of N=32 to 128 oscillators appear to be large enough to suggest statistical mechanical behavior. A key element has been a comparison of the maximum Lyapunov coefficient ${\ensuremath{\lambda}}_{\mathrm{max}}$ of the FPU \ensuremath{\alpha} model and that of the Toda lattice. For generic initial conditions, ${\ensuremath{\lambda}}_{\mathrm{max}}$(t) is indistinguishable for the two models up to times that increase with decreasing energy (at fixed N). Then suddenly a bifurcation appears, which can be discussed in relation to the breakup of regular, solitonlike structures. After this bifurcation, the ${\ensuremath{\lambda}}_{\mathrm{max}}$ of the FPU model appears to approach a constant, while the ${\ensuremath{\lambda}}_{\mathrm{max}}$ of the Toda lattice appears to approach zero, consistent with its integrability. This suggests that for generic initial conditions the FPU \ensuremath{\alpha} model is chaotic and will therefore approach equilibrium and equipartition of energy. There is, however, a threshold energy density ${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{c}}$(N)\ensuremath{\sim}1/${\mathrm{N}}^{2}$, below which trapping occurs; here the dynamics appears to be regular, solitonlike, and the approach to equilibrium\char22{}if any\char22{}takes longer than observable on any available computer. Above this threshold the system appears to behave in accordance with statistical mechanics, exhibiting an approach to equilibrium in physically reasonable times. The initial conditions chosen by Fermi, Pasta, and Ulam were not generic and below threshold and would have required possibly an infinite time to reach equilibrium. The picture obtained on the basis of ${\ensuremath{\lambda}}_{\mathrm{max}}$ suggests that neither the KAM nor the Nekhoroshev theorems in their present form are directly relevant for a discussion of the phenomenology of the FPU \ensuremath{\alpha} model presented here.