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Equitable Access Policy

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Adjoint-Based Projections for Quantifying Statistical Covariance near Stochastically Perturbed Limit Cycles and Tori
(SIAM, 2025-05) Dankowicz, Harry; Ahsan, Zaid; Kuehn, Christian
This paper presents a new boundary-value problem formulation for quantifying uncertainty induced by the presence of small Brownian noise near normally hyperbolic attracting periodic orbits (limit cycles) and quasiperiodic invariant tori of the deterministic dynamical systems obtained in the absence of noise. The formulation uses adjoints to construct a continuous family of transversal hyperplanes that are invariant under the linearized deterministic flow near the limit cycle or quasiperiodic invariant torus. The intersections with each hyperplane of stochastic trajectories that remain near the deterministic cycle or torus over intermediate times may be approximated by a Gaussian distribution whose covariance matrix can be obtained from the solution to the corresponding boundary-value problem. In the case of limit cycles, the analysis improves upon results in the literature through the explicit use of state-space projections, transversality constraints, and symmetry-breaking parameters that ensure uniqueness of the solution despite the lack of hyperbolicity along the limit cycle. These same innovations are then generalized to the case of a quasiperiodic invariant torus of arbitrary dimension. In each case, a closed-form solution to the covariance boundary-value problem is found in terms of a convergent series. The methodology is validated against the results of numerical integration for two examples of stochastically perturbed limit cycles and one example of a stochastically perturbed two-dimensional quasiperiodic invariant torus in $\mathbb{R}^2$, $\mathbb{R}^2\times S^1$, and $\mathbb{R}^2\times S^1$, respectively, for which explicit expressions may be found for the associated covariance functions using the proposed series solutions. Finally, an implementation of the covariance boundary-value problem in the numerical continuation package \textsc{coco} is applied to analyze the small-noise limit near a two-dimensional quasiperiodic invariant torus in a nonlinear deterministic dynamical system in $\mathbb{R}^4$ that does not support closed-form analysis. Excellent agreement with numerical evidence from stochastic time integration shows the potential for using deterministic continuation techniques to study the influence of stochastic perturbations for both autonomous and periodically excited deterministic vector fields.
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Affordable Real-Time CV for Disaster Relief and Beyond
(2025-05) Linsider, Eitan; Rana, Jay; Roy, Nathan; Stern, Jacob; Deane, Anil E.
Search and rescue operations following natural disasters are critical for saving the lives of individuals trapped and in need of aid. However, many of the technologies used in search and rescue (helicopters, planes, boats, etc.) can be prohibitively expensive for nations with low GDP. This issue is only exacerbated by the higher number of deaths due to natural disasters in such nations due to less resilient infrastructure. Unmanned Aerial Vehicles (UAVs) have the potential to replace more expensive technologies in the search for endangered individuals and analysis of at-risk hardware without endangering rescue personnel. While advancements in low-cost UAVs for commercial and hobby use in addition to the development of lightweight computer vision (CV) software and dedicated processors have set the stage for low-cost search and rescue UAVs, stand-alone UAV costs and part scarcity still pose challenges. Team ANDRR developed a real-time CV module for UAVs to aid in natural disaster relief using a wide range of compatible commercially available parts wherever possible. By conglomerating existing systems into a single effective and easy-to-use unit, we were able to identify individuals in real-time at low cost. The development guide and software for the module are provided to provide increased access to real-time CV processing for search and rescue and other operations.