Stabilityremain

Stabilityremain is a metric used in the analysis of dynamical systems to quantify how long a system remains within a predefined stability region after a disturbance. In this framing, the stability region is the set of states considered acceptable or safe, based on a Lyapunov bound, state constraints, or performance criteria. The stabilityremain is the time from the disturbance event to the first exit time from that region. In stochastic settings, researchers may report the expected stabilityremain or its distribution.

Variants include average stabilityremain, worst-case stabilityremain, and probabilistic stabilityremain, depending on whether the analysis uses deterministic trajectories, ensembles, or probabilistic disturbances.

Computation typically relies on time-domain simulation to observe trajectories, reachability analysis to determine exit surfaces, or Lyapunov-based estimates that bound the exit time without full simulation.

Applications are described in control design, power systems, robotics, and ecological or financial models where a controlled or natural system must operate within limits for a required duration after a perturbation. The concept is used to assess resilience and operational risk by focusing on the temporal persistence of stability rather than solely on perturbation tolerance or immediate return to equilibrium.

Relationship to related concepts: stabilityremain is related to, but distinct from, stability margin, which measures robustness of stability to parameter changes, and basin stability, which characterizes the likelihood of return to equilibrium. Stabilityremain emphasizes the temporal persistence within the stability region after a perturbation and can be adapted to deterministic or stochastic frameworks.