Same worst case, different risk: how transient response shapes failure probability
Abstract
The smallest disturbance that can push a dynamical system into failure is a standard measure of stability, but it does not determine how likely failure is under randomly directed disturbances of a given size. The geometry of the transient response connects the two. In the linearized dynamics, each failure boundary and each time after the disturbance define a response vector along the most damaging disturbance. The convex hull of these vectors, the response hull, determines the linear failure-probability curve through its support function. The longest vector sets the worst-case threshold. The mean and spread of the support function set where the curve rises and how steeply, but the mean does not rank systems by risk in general. Under mild regularity conditions, an exact linear-response relation shows how a parameter change shifts the disturbance size at which a given fraction fails: through the mean susceptibility of the directions on that threshold. For weakly damped oscillators, the susceptibility to damping is approximately proportional to the time of a direction's dominant response. Damping thus acts mainly on late failure routes rather than on the early one that sets the worst case, changing the failure probability substantially while barely moving the worst-case threshold. At fixed power imbalance, lowering the inertia instead nearly rescales the hull, moving worst case and risk together. In benchmark power-grid models, the linear predictions agree closely with nonlinear simulations. Strong nonlinear coupling, invisible to linear response, can break this picture; we show how it enters at second order.
Related
- Comparison of Different Models in Calculation of Tumor Control Probability and Normal Tissue Complication Probability in Breast Cancer Radiotherapy.
- On the Failure Probability of Quantum Leftover Hashing with Bad Seeds
- Thermal Inverse Design: Tailoring the Transient Response of Resistive Micro-Heaters
- Response of GAGG(Ce) scintillators to different charged particles
- The Measure Problem for Observers: Probability, Typicality, and Self-Location Without a Global Sampler
BibTeX
@misc{limkumnerd2026same,
title = {{Same worst case, different risk: how transient response shapes failure probability}},
author = {Surachate Limkumnerd},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.00789},
eprint = {2606.00789},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.00789},
}
RIS
TY - UNPB
TI - Same worst case, different risk: how transient response shapes failure probability
AU - Surachate Limkumnerd
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2606.00789
UR - https://arxiv.org/abs/2606.00789
AB - The smallest disturbance that can push a dynamical system into failure is a standard measure of stability, but it does not determine how likely failure is under randomly directed disturbances of a given size. The geometry of the transient response connects the two. In the linearized dynamics, each failure boundary and each time after the disturbance define a response vector along the most damaging disturbance. The convex hull of these vectors, the response hull, determines the linear failure-probability curve through its support function. The longest vector sets the worst-case threshold. The mean and spread of the support function set where the curve rises and how steeply, but the mean does not rank systems by risk in general. Under mild regularity conditions, an exact linear-response relation shows how a parameter change shifts the disturbance size at which a given fraction fails: through the mean susceptibility of the directions on that threshold. For weakly damped oscillators, the susceptibility to damping is approximately proportional to the time of a direction's dominant response. Damping thus acts mainly on late failure routes rather than on the early one that sets the worst case, changing the failure probability substantially while barely moving the worst-case threshold. At fixed power imbalance, lowering the inertia instead nearly rescales the hull, moving worst case and risk together. In benchmark power-grid models, the linear predictions agree closely with nonlinear simulations. Strong nonlinear coupling, invisible to linear response, can break this picture; we show how it enters at second order.
ER -