Abstract:

The UPC dynamical systems group is composed by 20 full time doctors and 3 predocs. Its research activity is addressed to the development of analytical and numerical methods for dynamical systems, holding a scientific leading position in several subjects, namely integrability, invariant objects, KAM theory, planar vector fields and applications to neuroscience, Poincaré-Melnikov method, exponentially small splitting of separatrices, non-smooth systems, celestial mechanics and astrodynamics. The quality of this activity is reflected in the quantity and quality of the publications, some of them in collaboration with cutting edge international research groups as well as in scientific management, leading to the organization of several events (conferences and meetings, research programs, themed networks, etc), the participation in several editorial boards of highly regarded journals and a large number of doctoral students.]]>

Abstract:

This project belongs to the area of Dynamical Systems and Applications, and it is the natural continuation of projects MTM2006-00478, MTM2009-06973 and MTM2012-31714, focusing on the local study and, mainly, the global study of continuous and discrete dynamical systems using analytical and numerical tools. The Dynamical Systems Group at the UPC is broad and interdisciplinary, combining experienced researchers with young researchers. This high research potential, coupled with a solid theoretical base, makes it possible to cover a wide set of classic and new problems in Dynamical Systems. The goal of the project consists on keep our leadership in the areas of Arnold Diffusion, splitting of separatrices and Astrodynamics, as well as advance in the study of bifurcations, computation of invariant objects and integrability. But we also want to strengthen the application of Dynamical Systems tools to problems in Astrodynamics, Celestial Mechanics, Chemistry and, as was most recently done with great success, in infinite dimensional Dynamical Systems and Neuroscience. In order to obtain numerical results not only in some theoretical problems, like exponentially small splitting of separatrices, but also in applications to Astrodynamics, Celestial Mechanics or Neuroscience, a highly computational approach is necessary, and very often it is only possible through parallel computation (the group owns and maintains a HPC cluster). The group already has a widely recognized expertise and prestige in these fields, and we plan to continue working on them in the following years Finally, it is important to stress that the group seeks a balance between working on issues where it is already expert and internationally recognized, and do it in new and more challenging problems for the group, such as infinite dimensional Dynamical Systems and Neuroscience. Our goals for the next three years are the following: A. Arnold Diffusion A.1. A-priori unstable Systems. A.2. A-priori stable Systems. A.3. Applications. B. Exponentially small phenomena B.1. Splitting of separatrices to low dimensional invariant objects. B.2. Splitting of separatrices in the Hopf-zero singularity. B.3. Using Melnikov potential in the problem of splitting of separatrices. B.4. Billiards next to the boundary. C. Integrability C.1. Planar polinomial vector fields. C.2. Algebraic integrable systems. D. Invariant objects and their bifurcations D.1. Invariant manifold of parabolic objects. D.2. Filippov Systems. D.3.Planar reversible diffeomorphisms. D.4. Planar vector fields. D.5. Quasi- periodic dynamics. D.6. Applications. E. Infinite Dimensional Dynamical Systems. E.1. Growth of Sobolev norms in Hamiltonian PDEs. E.2.Singular problems in PDEs. F. Astrodynamics F.1. Generation of natural and artificial trajectories. F.2. Artificial satellite formation. F.3. Analysis of orbits in the three body problem. G. Neuroscience and Biomedical applications G.1. Dynamics of models for single neurons and neural populations. G.2. Biomedical applications.]]>