A Moment-Based Eulerian Method for Variance-Based Finite-Time Lyapunov Exponent Computation in Stochastic Flows

Mathematics > Dynamical Systems [Submitted on 18 Jun 2026] Title:A Moment-Based Eulerian Method for Variance-Based Finite-Time Lyapunov Exponent Computation in Stochastic Flows View PDF HTML (experimental)Abstract:Variance-based finite-time Lyapunov exponents (vFTLEs) provide a stochastic analogue of deterministic FTLE by measuring the covariance of stochastic arrival locations. Existing PDF-based formulations compute this covariance by solving a Fokker--Planck equation for each initial point, which becomes expensive when the diagnostic is required on a dense grid. In this work, we develop a moment-based Eulerian approximation to vFTLE in the small-noise regime. Starting from a stochastic trajectory expansion about the deterministic flow, we derive a closed covariance equation for the leading stochastic displacement. By embedding this trajectory-wise covariance dynamics into physical space, we obtain an Eulerian transport--reaction equation for a symmetric covariance tensor field. The covariance associated with each initial point is recovered by evaluating this tensor field at the deterministic arrival location, and a moment-based vFTLE is then defined from its largest eigenvalue. The proposed method replaces a family of Fokker--Planck solves by the evolution of a single covariance tensor field, requiring only $d(d+1)/2$ scalar fields in $d$ dimensions. It also retains directional information through the eigenvectors of the covariance tensor, allowing the dominant directions of stochastic spreading to be visualized. We establish the leading-order consistency of the method with PDF-based vFTLE in the small-noise limit, clarify its relation to scalar stochastic sensitivity, and show how the same covariance equation connects process-noise spreading with deterministic deformation. In particular, deterministic FTLE is recovered, up to an additive constant, from an isotropic initial covariance when no process noise is present, while continuous process noise produces a time-integrated deformation covariance. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.