Rigorous computation for infinite dimensional nonlinear dynamics

This workshop, sponsored by AIM and the NSF, will focus on the extension of the rigorous computational tools used in finite dimensional dynamical systems to the infinite case.

The existence of chaotic dynamics was first demonstrated by Poincare more than a century ago. Its relevance to science and engineering exploded 50 years ago with the advent of the computer and the ability to simulate concrete nonlinear systems. This in turn stimulated the development of the very rich and beautiful theory of dynamical systems. And yet, given a particular nonlinear system of differential equations or a particular nonlinear map any comprehensive understanding of the associated dynamics is typically obtained through numerical methods which are not rigorous and may be misleading in some critical cases of particular interest. Motivated in part by this quandary the past few decades have seen substantial advances in the development of computer assisted proofs in finite dimensional dynamics. The majority of these techniques are based on either the contraction mapping theorem or algebraic topology. The choice of strategy has profound implications on the numerical methods that should be employed.

This workshop will focus on the extension of these rigorous computational tools to infinite dimensional dynamical system (in particular evolutionary partial differential equations). The following general questions will be used to frame the technical challenges facing this project.

• Where do our current algorithms fail in the infinite dimensional setting and how should we proceed to improve them?
• Which classes of infinite dimensional systems are most amenable to current methods? What needs to be done to expand the classes?
• What types of dynamical structures of phenomena, e.g. fixed points, periodic orbits, connecting orbits, invariant manifolds, bifurcations, can be verified using rigorous computing in the near future? What needs to be done to expand these types?
• Are there problems that can serve as grand challenges to focus the development of these methods?

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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