A proposal has been approved by the Lorentz Center, Leiden, The Netherlands, to carry out a workshop entitled ?Coherent structures in dynamical systems? on 16 May 2011 - 20 May 2011. This NSF support will partially cover the travel cost of US participants in this workshop. The organization of the workshop and substantial part of the cost are provided by the Lorentz Center (www.lorentzcenter.nl) .
The workshop focus on new collaborations between scientists from different countries and fields, bringing together groups of 30 to 40 junior and senior researchers in a stimulating environment with working space for all participants. Through a combination of informal talks, working sessions, and discussions, participants are able to assess the status of a field and its future, and to collaborate, establish new international contacts, and spot upcoming talent.
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. 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 the workshop, therefore, is to bring together theoretical experts and researchers with interests in the application of dynamical systems methods. Such a gathering is expected to inspire new theoretical research and new areas of application. The workshop will build on the success of the recent Focus Issue on "Lagrangian Coherent Structures in Fluid Flows" in the journal Chaos, which was featured in The New York Times and The Economist. Developing robust methods for uncovering coherent structures in fluid systems will directly influence our ability to make reliable predictions for such high-profile scenarios as the Deepwater Horizon oil spill and the Icelandic volcanic ash cloud, both of which have had tremendous societal impact.
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 the above analyses is only feasible when explicit knowledge of the dynamical system equations 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, such the ocean and the Earth's and planetary atmospheres. 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. Knowledge of such a skeleton is critically important, for instance, to both understand and predict the motion of pollutants such as oil spills in the ocean or volcanic ashes in the atmosphere. 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.
