| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | April 3, 2006 |
| Latest Amendment Date: | March 27, 2010 |
| Award Number: | 0600974 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
Bruce P. Palka
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | June 1, 2006 |
| End Date: | May 31, 2012 (Estimated) |
| Total Intended Award Amount: | $575,000.00 |
| Total Awarded Amount to Date: | $575,000.00 |
| Funds Obligated to Date: |
FY 2007 = $115,000.00 FY 2008 = $115,000.00 FY 2009 = $115,000.00 FY 2010 = $115,000.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
70 WASHINGTON SQ S NEW YORK NY US 10012-1019 (212)998-2121 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
New York University New York NY US 10012-1092 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | ANALYSIS PROGRAM |
| Primary Program Source: |
app-0107 01000809DB NSF RESEARCH & RELATED ACTIVIT 01000910DB NSF RESEARCH & RELATED ACTIVIT 01001011DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
Research topics within the theory of dynamical systems and on its applications are proposed. Four projects are described. The first proposes to leverage techniques developed for low dimensional systems to analyze (infinite dimensional) systems defined by evolutionary partial differential equations. Expected results include methods for detecting strange attractors in physical and mechanical systems. The second project is about networks of dynamical systems. It seeks to systematically relate the aggregate behaviors of such systems to those of their individual components and to the coupling. The third topic lies in the interface between dynamical systems and nonequilibrium statistical physics. One of the objectives here is to shed light on the notion of local equilibrium for systems with deterministic microscopic dynamics. The fourth and final topic is about large deviations, an important statistical property for dynamical systems. A scheme expected to resolve the issue for a large class of nonuniformly hyperbolic systems is proposed.
This proposal addresses several topics at the frontier of research in dynamical systems, a branch of modern mathematics concerned with time evolutions of natural processes. Its main focus is the analysis of systems with high complexity, due either to multiple degrees of freedom or to chaos within the system. It is systems of this kind that are most often encountered in applications. The proposed methods of investigation include geometric analysis, probabilistic techniques and numerical simulations. A larger aim of the proposed work is to integrate dynamical systems ideas into other core areas of mathematics such as partial differential equations, and to build connections with other scientific disciplines such as statistical physics and the biological sciences. The resulting cross-fertilization is expected to be beneficial to all. In terms of educational value, the proposed research will provide ample training for emerging mathematicians, as it will involve directly the postdoctoral associates and Ph.D. students of the Principle Investigator.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
The research conducted was on dynamical systems, a branch of modern mathematics concerned with the time evolution of processes. Four major steps were taken during the period of the funding. The first topic concerns strange attractors of a particular type. While ubiquitous in nature, these objects had resisted mathematical analysis for quite some time. The main thrust of the PI's work was to identify checkable conditions that imply (indirectly) the presence of these attractors. As a result of her new tools, the set of strange attractors amenable to mathematical analysis was greatly expanded. The second topic of this research clarified certain relations between dynamical systems and probabilistic processes. Specifically, it shed light on how chaotic behavior can lead to uncertainties, even when the rules governing the dynamics are precise and deterministic. The third and fourth topics are concerned with applications of dynamical systems ideas to other scientific disciplines. The third topic lies in the interface between dynamical systems and nonequilibrium statistical mechanics. The PI used mathematical models to study fundamental theoretical questions regarding heat conduction. The fourth project is on networks of coupled oscillators. The PI worked with models built to simulate neuronal networks, focusing on the question of reliability, or when a system can be counted on to respond consistently in the face of multiple ongoing processes with possibly conflicting demands.
With regard to broader impact, the research carried out under this grant served two purposes: it both advanced current mathematical theory of chaotic dynamical systems, and brought basic dynamical systems theories closer to their potential applications. Such applications include various aspects of physical, biological and engineered systems.
Last Modified: 08/30/2012
Modified by: Lai-Sang Young
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