  We study the onset of chaos and statistical relaxation in two isolated
dynamical quantum systems of interacting spins-1/2, one of which is integrable
and the other chaotic. Our approach to identifying the emergence of chaos is
based on the level of delocalization of the eigenstates with respect to the
energy shell, the latter being determined by the interaction strength between
particles or quasi-particles. We also discuss how the onset of chaos may be
anticipated by a careful analysis of the Hamiltonian matrices, even before
diagonalization. We find that despite differences between the two models, their
relaxation process following a quench is very similar and can be described
analytically with a theory previously developed for systems with two-body
random interactions. Our results imply that global features of statistical
relaxation depend on the degree of spread of the eigenstates within the energy
shell and may happen to both integrable and non-integrable systems.
