  In network systems, a local perturbation can amplify as it propagates,
potentially leading to a large-scale cascading failure. Here we derive a
continuous model to advance our understanding of cascading failures in
power-grid networks. The model accounts for both the failure of transmission
lines and the desynchronization of power generators, and incorporates the
transient dynamics between successive steps of the cascade. In this framework,
we show that a cascade event is a phase-space transition from an equilibrium
state with high energy to an equilibrium state with lower energy, which can be
suitably described in closed form using a global Hamiltonian-like function.
From this function we show that a perturbed system cannot always reach the
equilibrium state predicted by quasi-steady-state cascade models, which would
correspond to a reduced number of failures, and may instead undergo a larger
cascade. We also show that in the presence of two or more perturbations, the
outcome depends strongly on the order and timing of the individual
perturbations. These results offer new insights into the current understanding
of cascading dynamics, with potential implications for control interventions.
