  We investigate the long term evolution of trajectories in the
Fermi-Pasta-Ulam (FPU) system, using as a probe the first non--trivial integral
$J$ in the hierarchy of integrals of the corresponding Toda lattice model. To
this end we perform simulations of FPU--trajectories for various classes of
initial conditions produced by the excitation of isolated modes, packets, as
well as `generic' (random) initial data. For initial conditions corresponding
to localized energy excitations, $J$ exhibits variations yielding `sigmoid'
curves similar to observables used in literature, e.g. the `spectral entropy'
or various types of `correlation functions'. However, $J(t)$ is free of
fluctuations inherent in such observables, hence it constitutes an ideal
observable for probing the timescales involved in the stages of FPU dynamics.
We observe two fundamental timescales: i) the `time of stability' (in which,
roughly, FPU trajectories behave like Toda), and ii) the `time to equilibrium'
(beyond which energy equipartition is reached). Below a specific energy
crossover, both times are found to scale exponentially as an inverse power of
the specific energy. However, this crossover goes to zero with increasing the
degrees of freedom $N$ as $\varepsilon _c \sim N^{-b}$, with $b \in [1.5,
2.5]$. For `generic data' initial conditions, instead, $J(t)$ allows to
quantify the continuous in time slow diffusion of the FPU trajectories in a
direction transverse to the Toda tori.
