  From mass extinction to cell death, complex networked systems often exhibit
abrupt dynamic transitions between desirable and undesirable states. Such
transitions are often caused by topological perturbations, such as node or link
removal, or decreasing link strengths. The problem is that reversing the
topological damage, namely retrieving the lost nodes or links, or reinforcing
the weakened interactions, does not guarantee the spontaneous recovery to the
desired functional state. Indeed, many of the relevant systems exhibit a
hysteresis phenomenon, remaining in the dysfunctional state, despite
reconstructing their damaged topology. To address this challenge, we develop a
two-step recovery scheme: first - topological reconstruction to the point where
the system can be revived, then dynamic interventions, to reignite the system's
lost functionality. Applying this method to a range of nonlinear network
dynamics, we identify the recoverable phase of a complex system, a state in
which the system can be reignited by microscopic interventions, for instance,
controlling just a single node. Mapping the boundaries of this dynamical phase,
we obtain guidelines for our two-step recovery.
