Abstract:

This project belongs to the area of Dynamical Systems and Applications, and follows the research lines of projects MTM2009-06973, MTM2012-31714,and MTM2015-65715-P, focusing on the local study and, mainly, the global study of continuous and discrete Dynamical Systems using analytical and numerical tools. The research team of the project is broad and interdisciplinary, combining experienced researchers with young researchers. Nevertheless, we would like to increase the quantity of young people in the team. This strong research potential, jointly with a solid theoretical base, makes it possible to cover a wide range of classic and new problems in Dynamical Systems. The size of the team in previous projects and the diversification of the research topics pushed us to split and to present two different proposals in this call. The present project (whose research team has 13 researchers) puts strong emphasis on the development of mathematical theory while aims to boost the applications to Mechanics and, especially, to Neuroscience and Mathematical Biology. To pursue this goal we have incorporated a junior co-PI with exceptional experience in Dynamical Systems and mathematical neuroscience. We want to on keep our leadership in the areas of Arnold Diffusion and splitting of separatrices, as well as to advance in the study of bifurcations, computation of invariant objects and integrability. But we also want to strengthen the applications of Dynamical Systems tools to problems in infinite dimensional Dynamical Systems (partial differential equations) and mathematical Neuroscience, where the group has already done important contributions. In order to obtain numerical results, not only in some theoretical problems, like exponentially small splitting of separatrices, but also in applications to Neuroscience, we follow a highly computational approach, 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 topics for which it is already expert and internationally recognized, like Arnold diffusion and splitting of separatrices, and working on new and challenging problems for the group, such as infinite dimensional Dynamical Systems and Neuroscience. We now list the main goals for the next 3 years. A more detailed subdivision in subprojects with more specific goals can be found in the research proposal. A. Arnold diffusion B. Exponentially small phenomena C. Integrablility D. Invariant objects and their bifurcations E. Infinite-dimensional Dynamical Systems F. Neuroscience and Mathematical Biology]]>

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.]]>