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Coherent Structures in Dynamical Systems
Central in the study of dynamical systems is the search for invariant sets or coherent structures that organize long-term behavior. Examples of coherent structures include periodic orbits, invariant manifolds, homoclinic orbits, and invariant tori. By performing appropriate local analyses around these coherent structures and studying how they fit together, a “skeleton” of the dynamics may be sketched. The specific characteristics of such a skeleton depend on the nature of the vector field at hand. Performing these analyses is only feasible when explicit knowledge of the vector field is available. But in many important problems such knowledge is not available, presenting a great challenge. An example is the motion of elementary fluid particles in a geophysical fluid. The time-aperiodic character of the mostly-turbulent velocity field, in addition to the finiteness of the time interval of the data, further complicates the possibility of drawing a skeleton of the Lagrangian dynamics.
In recent years, new techniques have been devised to help unveil dynamical skeletons from limited data sets. These techniques have been mainly applied to unsteady flows in two-space dimensions, leading to a major breakthrough in the physics of mixing; their application to unsteady three-dimensional flows remains largely unexplored. For instance, it remains to be determined if there exists a rigorous theory that can be used as a foundation for coherent structures underlying such flows. An accompanying challenge is the appropriate visualization of coherent structures evolving in three-space dimensions and time.
The aim of this workshop was to bring together experts on the study of coherent structures in dynamical systems, and researchers with interests in the application of dynamical systems methods to unveil coherent structures in dynamical systems that describe specific physical systems including, but not restricted to, fluid systems.
As a result of this gathering it was expected that applied dynamical systems researchers benefit from the expertise of theoretical dynamical systems researchers. At the same time it was expected that new theoretical research in dynamical systems be inspired by specific applications of dynamical systems methods. This workshop was thought to be considered a success if, in addition to promoting the collaboration between applied and theoretical dynamical systems researchers, young researchers got interested in this new and exciting area.
New developments in theoretical dynamical systems research and results from the application of dynamical systems tools were presented in the workshop by both senior investigators and young researchers, including a good number of graduate students. Some of the new theoretical developments were inspired by needs of advancing applied research, mainly in the area of fluid dynamics. Additional needs of advancing theoretical research were brought to the attention to the theoretical community, which are expected to stimulate further interactions among theoretical and applied researchers that lead to new advancements in the study of coherent structures in dynamical systems. Overall, it can be confidently stated that the goals of the workshop have been reached.