This research proposal is mainly on the subject of non-uniformly hyperbolic strange attractors. We propose to apply an abstract setting we previously formulated and studied to rigorously prove the existence of non-uniformly hyperbolic strange attractors in certain systems of ordinary differential equations, including a modified system of van der Pol and certain periodically excited systems experiencing Hopf bifurcations. We also propose to analytically establish the fact that non-uniformly hyperbolic strange attractors naturally arise when an asymptotically stable periodic solution of a given system of ordinary differential equations is periodically excited by generic external forcing. These non-uniformly hyperbolic strange attractors have properties that include most of standard mathematical notions associated with chaos: positive Liapunov exponents, positive entropy, SRB measure, exponential decay of correlation, nice symbolic coding of orbits etc. We also propose to study the geometric structure of the integral manifold of the spatial four-body problem.

This research proposal is mainly on the subject of non-uniformly hyperbolic strange attractors. In general an attractor is a state to which a system will eventually evolve. One of the most important way of studying systems in nature is to first model then study them as solutions of certain differential equations. We propose to apply a theory we previously developed to prove the existence of a class of strange attractors in some systems of differential equations arisen from various scientific disciplines including in the studies of turbulence, fluid mechanics and plasma mechanics. The attracting states we propose to study have very complicated dynamical properties and sophisticated geometric structures. They were observed in many numerical and laboratory experiments but their existence were rarely established mathematically in the past. We also propose to study the geometric structure of the integral manifold of the spatial four-body problem. This is a long standing mathematical problem with potential applications on orbit design for artificial celestial objects.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
0204725
Program Officer
Christopher W. Stark
Project Start
Project End
Budget Start
2002-07-01
Budget End
2005-06-30
Support Year
Fiscal Year
2002
Total Cost
$78,072
Indirect Cost
Name
University of Arizona
Department
Type
DUNS #
City
Tucson
State
AZ
Country
United States
Zip Code
85721