Bounding extreme events in dynamical systems over an infinite time horizon
Abstract
This work addresses the computation of extreme values of observables along trajectories of dynamical systems over an \ks{infinite time horizon}. The setting is intrinsically global: trajectories may exhibit large transient excursions, approach their largest values only asymptotically as time tends to infinity, or escape to infinity in finite time. The proposed framework treats these \mk{ scenarios} in a unified way, so that the resulting extreme values reflect both the transient and the asymptotic behavior of the dynamics. The main difficulty is that neither the time interval nor the state space is assumed to be compact. We address this by combining a radial compactification of the state space with a compactified clock variable and discounted occupation measures. This yields a description of trajectory ensembles on compact sets via occupation measures and the associated Liouville's equation. Extreme values are then characterized through a family of reachability problems for level sets of the observable, which can be solved by a bisection procedure. On the theoretical side, we prove a superposition principle for an infinite-horizon Liouville's equation with a time-distributed terminal measure. This result shows that the measure formulation represents ensembles of stopped admissible trajectories, including trajectories \ks{that never stop}, and establishes equivalence between the trajectory-level and measure-level descriptions.
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BibTeX
@misc{sehnalova2026bounding,
title = {{Bounding extreme events in dynamical systems over an infinite time horizon}},
author = {Karolı́na Sehnalová and Milan Korda},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30821},
eprint = {2609.30821},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05761573v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bounding extreme events in dynamical systems over an infinite time horizon
AU - Karolı́na Sehnalová
AU - Milan Korda
PY - 2026
DA - 2026/09/23
PB - HAL
DO - 10.48550/arxiv.2609.30821
UR - https://hal.science/hal-05761573v1
AB - This work addresses the computation of extreme values of observables along trajectories of dynamical systems over an \ks{infinite time horizon}. The setting is intrinsically global: trajectories may exhibit large transient excursions, approach their largest values only asymptotically as time tends to infinity, or escape to infinity in finite time. The proposed framework treats these \mk{ scenarios} in a unified way, so that the resulting extreme values reflect both the transient and the asymptotic behavior of the dynamics. The main difficulty is that neither the time interval nor the state space is assumed to be compact. We address this by combining a radial compactification of the state space with a compactified clock variable and discounted occupation measures. This yields a description of trajectory ensembles on compact sets via occupation measures and the associated Liouville's equation. Extreme values are then characterized through a family of reachability problems for level sets of the observable, which can be solved by a bisection procedure. On the theoretical side, we prove a superposition principle for an infinite-horizon Liouville's equation with a time-distributed terminal measure. This result shows that the measure formulation represents ensembles of stopped admissible trajectories, including trajectories \ks{that never stop}, and establishes equivalence between the trajectory-level and measure-level descriptions.
ER -