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The Fermi-Pasta-Ulam α-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 Lyapounov coefficient λ max of the FPU α-model and that of the Toda lattice. For generic initial conditions, λ 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, soliton-like structures. After this bifurcation, the λ max of the FPU model appears to approach a constant, while the λ max of the Toda lattice appears to approach zero, consistent with its integrability.This suggests that for generic initial conditions the FPU α-model is chaotic and will therefore approach equilibrium and equipartition of energy. There is, however, a threshold energy density ǫ c (N ) ∼ 1/N 2 , below which trapping 1 occurs; here the dynamics appears to be regular, soliton-like and the approach to equilibrium -if any -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 rea-

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