  In a system of interdependent networks, an initial failure of nodes invokes a
cascade of iterative failures that may lead to a total collapse of the whole
system in a form of an abrupt first order transition. When the fraction of
initial failed nodes $1-p$ reaches criticality, $p=p_c$, the abrupt collapse
occurs by spontaneous cascading failures. At this stage, the giant component
decreases slowly in a plateau form and the number of iterations in the cascade,
$\tau$, diverges. The origin of this plateau and its increasing with the size
of the system remained unclear. Here we find that simultaneously with the
abrupt first order transition a spontaneous second order percolation occurs
during the cascade of iterative failures. This sheds light on the origin of the
plateau and on how its length scales with the size of the system. Understanding
the critical nature of the dynamical process of cascading failures may be
useful for designing strategies for preventing and mitigating catastrophic
collapses.
