The main goal of this proposal is to devise methods that allow to make predictions of the long term (or the long range) behavior of dynamical systems or PDE. We plan to develop a broad array of tools (invariant manifolds, variational methods, numerical analysis) in such a way that they can work together. We are particularly interested in applications to instability in dynamical systems and to global behavior in elliptic partial differential equations and in coupled networks.
Many of the laws of nature are formulated as local interactions. One point in space and time affects only its close neighborhood. It can happen that these local interactions cancel each other out so that the global effect is small and that the systems remain kind of unaffected or it can happen that the local interactions reinforce each other and lead to large scale effects. The two alternatives do happen and they depend on very subtle effects (e.g. rather deep and abstract number theory is the key to very measurable effects). Even if the importance of the has been recognized by applied mathematicians for centuries, it is only very recently that a rich enough toolkit has been developed by many start tackling it. Different people, have been making different techniques to work together, and they have started producing results. As a witness to the interest, the PI of this proposal has been co-organizer of special semesters in CRM (Barcelona) Fall 2008 and Fields institute (Spring 2011).
It is well known that even simple rules of evolution can lead to complicated behaviors when applied repeatedly. This happens even in practical systems, for example in Newton's model of the solar System. The goal of this project was to develop both rigorous mathematical theorems and numerical tools that allow to get a handle on these complicated behaviors. The main idea is to identify some special motions which are very easy to predict. These act as landmarks that organize the complicated chaotic behavior. The mathematical theorems jiustify the reliability of the methods and, on the other hand, the numerical explorations suggest new conjectures and more theorems. The techniques used are sometimes surprising since one has to take into account how well can one approximate the frequencies by quotients of integer numbers keeping the numerators smaller. Once these landmarks are identified (and computed) they can serve as the skeleton for more complicated behaviors that can be classified or exploited for practical purposes. We paid specia attention to models coming from astrodynamis and from solid state physics (variations on the Frenkel-Kontorova model for deposition of solids and also the coupled oscillators used in neurosciences. One application considered was suggested by chemists trying to identify mechanisms of reations. In the process of pursuing these scientific goals, the project trained several students and developed international collaborations.
