  We investigate the quench dynamics of a one-dimensional strongly chaotic
lattice with $L$ interacting spins. By analyzing both the classical and quantum
dynamics, we identify and elucidate the two mechanisms of the relaxation
process of these systems: one arises from linear parametric instability and the
other from nonlinearity. We demonstrate that the relaxation of the
single-particles energies (global quantity) and of the onsite magnetization
(local observable) is primarily due to the first mechanism, referred to as
linear chaos. Our analytical findings indicate that both quantities, in the
classical and quantum domain, relax at the same timescale, which is independent
of the system size. The physical explanation for this behavior lies in the
conservation of the $L$ spin angular momenta. We argue that observables with a
well-defined classical limit should conform to this picture and exhibit a
finite relaxation time in the thermodynamic limit. In contrast, the evolution
of the participation ratio, which measures how the initial state spreads in the
many-body Hilbert space and has no classical limit, indicates absence of
relaxation in the thermodynamic limit.
