  We propose a novel framework to characterize the thermalization of many-body
dynamical systems close to integrable limits using the scaling properties of
the full Lyapunov spectrum. We use a classical unitary map model to investigate
macroscopic weakly nonintegrable dynamics beyond the limits set by the KAM
regime. We perform our analysis in two fundamentally distinct long-range and
short-range integrable limits which stem from the type of nonintegrable
perturbations. Long-range limits result in a single parameter scaling of the
Lyapunov spectrum, with the inverse largest Lyapunov exponent being the only
diverging control parameter and the rescaled spectrum approaching an analytical
function. Short-range limits result in a dramatic slowing down of
thermalization which manifests through the rescaled Lyapunov spectrum
approaching a non-analytic function. An additional diverging length scale
controls the exponential suppression of all Lyapunov exponents relative to the
largest one.
