  The Fermi-Pasta-Ulam $\alpha$-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 Lyapounov coefficient $\lambda_{max}$ of the FPU $\alpha$-model and
that of the Toda lattice. For generic initial conditions, $\lambda_{max}(t)$ 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 $\lambda_{max}$ of the FPU model appears to approach a
constant, while the $\lambda_{max}$ of the Toda lattice appears to approach
zero, consistent with its integrability. This suggests that for generic initial
conditions the FPU $\alpha$-model is chaotic and will therefore approach
equilibrium and equipartition of energy. There is, however, a threshold energy
density $\epsilon_c(N)\sim 1/N^2$, below which trapping 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 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.
