  We compute the Lyapunov exponent characterizing quantum scrambling in a
family of generalized Sachdev-Ye-Kitaev models, which can be tuned between
different low temperature states from Fermi liquids, through non-Fermi liquids
to fast scramblers. The analytic calculation, controlled by a small coupling
constant and large $N$, allows us to clarify the relations between the
quasi-particle relaxation rate $1/\tau$ and the Lyapunov exponent $\lambda_L$
characterizing scrambling. In the Fermi liquid states we find that the
quasi-particle relaxation rate dictates the Lyapunov exponent. In non-Fermi
liquids, where $1/\tau \gg T$, we find that $\lambda_L$ is always $T$-linear
with a prefactor that is independent of the coupling constant in the limit of
weak coupling. Instead it is determined by a scaling exponent that
characterizes the relaxation rate. $\lambda_L$ approaches the general upper
bound $2\pi T$ at the transition to a fast scrambling state. Finally in a
marginal Fermi liquid state the exponent is linear in temperature with a
prefactor that vanishes as a non analytic function $\sim g \ln (1/g)$ of the
coupling constant $g$.
