Transformations for Geometric Analysis Reveal Simplicity of Complex Dynamical Systems
Abstract
PUBLICLY RELEASABLESignificance of the proposed work: Representation of a dynamical system in terms of simplifying modes is a centra, is to represent something complicated as something simple. It is the change of variables itself, to a simplified presentation that, uncovers the nature of the dynamical system since the maximal sets over which a transformation to the simplest of all forms, being,integrability, are related to domains of coherence. Also, changes like these are indicative of important changes like critical trans,itions, onset of crisis, or even precursors of extreme events. These are encoded in, 1) The geometry associated with transformation, to integrability, and 2) These transformations in turn are encoded by considering the set of level sets of eigenfunctions of transf,er operators. Directly studying these changes of variables is key to revealing the simplicity hidden in even chaotic motions. From, fluid dynamics including oceans and the atmosphere, to structural mechanics, to other nonlinear problems such as nonlinear optics,,biomedical applications and population dynamics, explaining complicated nonlinear behavior and perhaps chaotic dynamical systems off,er pressing and ongoing difficult issues. The premise of this work is that we offer a powerful way to understand these systems by g,eometric properties associated with the set of level sets of eigenfunctions of transfer operators. Since transformations to other s,ystems are encoded in the eigenfunctions compared between two systems, the relationships between the sets of level sets between the,matched sets of eigenfunctions characterizes the dynamics. Technical Approaches: When compared between complicated and simplified,forms, matched eigenfunctions of transfer operators will allow development of change of variables to integrable form. Analysis of t,he domains of these will characterize the complexity of the system.Analysis of geometry of the sets of level sets of primary eigenf,unctions of each system distinguishes domains in which the reduced order reduction may occur, will define the reduction.Important s,ystem properties such as transport or coherence are also,We will study simplified synthetic models with complete closed form solutions toward eigenfunctions and their level sets, and more,complex synthetic models with realistic complicated motions (e.g. double gyre, and Rossby wave system) to test numerical methods for, solving the eigenfunction quasi-linear PDE, continuation of solutions and topological features of level sets, and then full data-dr,iven methods for analysis (inverse problems methods for the operator and eigenfunctions from data) of interesting and relevant exper,imental or observational data.Anticipated Outcome of the Research:This work will reveal new methods for mathematical analysis of im,is work reveals that there is a great deal further of geometric information concerning global dynamics to be interpreted from the ma,nner in which the level sets of eigenfunctions intersect amongst themselves, and relate when considered between matched sets of syst,ems.This new kind of global analysis suggest a way to describe comprehensive properties such as transport, and coherence as well as, system changes such as onset of criticality, and tipping simply by associating geometric changes to typical forms. Impact on DoD C,apabilities:Complex systems that are specifically Naval and generally DoD relevant will benefit from a new and powerful analytic too,l. Fluidic systems including oceanographic and atmospheric systems will benefit from this new analysis, as well as engineering desi
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Mar 05, 2022
- Source ID
- N000142212173
Entities
People
- Erik Bollt
Organizations
- Clarkson University
- Office of Naval Research
- United States Navy