2021/01/25 by Sanjeeva Balasuriya, Erik M. Bollt, Balasuriya, Sanjeeva +2
Engineering · Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Artificial intelligence #Chaotic #Chaotic Dynamics (nlin.CD) #Class (philosophy) #Computer science #Dynamical Systems (math.DS) #Dynamical system (definition) #Dynamical systems theory #FOS: Mathematics #FOS: Physical sciences #Flow (mathematics) #Fluid Dynamics and Turbulent Flows #Geometry #Gravitational singularity #Lyapunov exponent #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Navier-Stokes equation solutions #Physics #Pure mathematics #Quantum chaos and dynamical systems #Sequence (biology) #Standard map #math.DS #nlin.CD
paper · pdf · doi:10.48550/arxiv.2101.10387
arxiv created 2021/01/25 · openalex publication_date 2021/01/25 · arxiv updated 2021/01/27 · openalex created_date 2022/07/25 · openalex updated_date 2026/08/04
Understanding instabilities in dynamical systems drives to the heart of\nmodern chaos theory, whether forecasting or attempting to control future\noutcomes. Instabilities in the sense of locally maximal stretching in maps is\nwell understood, and is connected to the concepts of Lyapunov\nexponents/vectors, Oseledec spaces and the Cauchy--Green tensor. In this paper,\nwe extend the concept to global optimization of stretching, as this forms a\nskeleton organizing the general instabilities. The `map' is general but\nincorporates the inevitability of finite-time as in any realistic application:\nit can be defined via a finite sequence of discrete maps, or a finite-time flow\nassociated with a continuous dynamical system. Limiting attention to\ntwo-dimensions, we formulate the global optimization problem as one over a\nrestricted class of foliations, and establish the foliations which both\nmaximize and minimize global stretching. A classification of nondegenerate\nsingularities of the foliations is obtained. Numerical issues in computing\noptimal foliations are examined, in particular insights into special curves\nalong which foliations appear to veer and/or do not cross, and foliation\nbehavior near singularities. Illustrations and validations of the results to\nthe H 'enon map, the double-gyre flow and the standard map are provided.\n