Orbit methods seek to understand dynamical systems by investigating the large-scale structure of their orbits through combinatorial, geometric, and statistical techniques. The principal investigator has developed a range of tools to explore such spaces of orbits and to study notions of equivalence of these structures that allow for measuring and controlling distortions or rearrangments of them. The goal of this program is to extend these orbit methods to as broad a perspective as possible and to apply them to answer interesting questions. These methods have a significant history in the study of systems that preserve probability measures. Two active new directions of work are extension of this theory to the almost continuous case, and the Cantor minimal theory. Both involve improving the regularity of any allowed isomorphisms beyond measurability, in the first case to be continuous with probability one and in the second actually to be continuous. Another broad attempt at extending these methods is to study probability measures diffused on the leaves of a foliation. The diffused measure gives a mass distribution on the leaves. One removes all conditions, though linking the dynamics and the measure. For one-dimensional leaves an ergodic theorem can be proved in this general context, and extending it to higher dimensions appears possible. This is a potentially very fruitful domain for extending orbit methods to quite general large-scale structures. A variety of other directions are pursued, including the beginnings of a collaboration in applied fluid dynamics that uses the metrics developed for measuring large-scale distortion of orbits to create a bag of tools for assessing the accuracy of numerical models.
Large arrays of data arise in many contexts, from genomics to digital imaging to geology. Similar arrays arise in a mathematical context as trajectories or orbits in the space of states of a dynamical system. Orbit methods involve notions of the size of distortion of such arrays or different ways of measuring how similar two such arrays are. One tunes the notion of distortion or distance to the context of the problem. There is the possibility of deep cross-fertilization between areas of engineering and applied science and abstract dynamics in this context. For example, the evolutionary distance between species can be measured in terms of how similar their genomes are, where one must tune what one means by similar to make biological sense. As another example, in compressing the data in a visual image one is perhaps willing to lose some precision so long as the picture is not significantly distorted. What does one mean by a significant distortion? As a third example, in modeling the meanderings of rivers one must establish what it would mean for a model to capture the real behavior of this complex phenomenon. Again, this can take the form of a thoughtfully tuned notion of closeness of patterns. One thrust of this project is to seek collaborations that look for such cross-fertilization of ideas. This has already begun in the areas of genomics and of fluid flow.
Many complex physical phenomena are modeled as dynamical systems, which model evolution as a set of rules describing how the state of a system at one time evolves from the states of the system at previous times. Often such rules are coded as differential equations whose complexity dictates that approximate computer solutions must be employed for practical use. This project tackled two aspects of the study and use of dynamical systems (necessitated by the unfortunate death of the original Principal Investigator D. Rudolph and subsequent appointment of D. Estep to continue management of the project). The first aspect of the project developed theoretical tools for analyzing dynamical systems by investigating the large-scale structure of the orbits or trajectories of particles. The goal was to determine equivalencies among different orbits, classifying orbits based on equivalencies, and then measuring and controlling distortions or re-arrangements of orbits. The project resulted in advances on a number of technical issues arising in the study of dynamical systems that have both measure and topological structure. The second aspect of the project was concerned with quantifying the effects of numerical simulation error and random error in data and parameters on information computed from complex dynamical systems and using complex dynamical system models to carry out physical and engineering analysis. The project focused on complex multiphysics, multiscale systems that present exceedingly difficult challenges for computational mathematics, engineering and science. On such problems, numerical simulation error is always significant so that accounting for its effects on computed information and devising computational strategies for efficiently computing information of a stated accuracy are a high priority in nearly all scientific and engineering applications. This component of the project also developed tools for using differential equations for determination of physical conditions in a system from experimental data and predicting future behavior of a system. This aspect of the project was heavily interdisciplinary, with close ties to both national research laboratories and private companies, on applications in energy, biology, ecology, chemistry, materials science, fluid flow, astrophysics, hurricane flooding, and oil recovery and pollution in the earth’s surface.
