Objective A fundamental problem in the study of dynamical systems is to ascertain whether the effect of a perturbation on an integrable Hamiltonian system accumulates over time and leads to a large effect (instability) or it averages out (stability). Instabilities in nearly integrable systems, usually called Arnold diffusion, take place along resonances and by means of aframework of partially hyperbolic invariant objects and their homoclinic and heteroclinic connections.The goal of this project is to develop new techniques, relying on the role of invariant manifolds in the global dynamics, to prove the existence of physically relevant instabilities and homoclinic phenomena in several problems in celestial mechanics and Hamiltonian Partial Differential Equations.The N body problem models the interaction of N puntual masses under gravitational force. Astronomers have deeply analyzed the role of resonances in this model. Nevertheless, mathematical results showing instabilities along them are rather scarce. I plan to develop a new theory to analyze the transversal intersection between invariant manifolds along mean motion and secular resonances to prove the existence of Arnold diffusion. I will also apply this theory to construct oscillatory motions.Several Partial Differential Equations such as the nonlinear Schrödinger, the Klein-Gordon and the wave equations can be seen as infinite dimensional Hamiltonian systems. Using dynamical systems techniques and understanding the role of invariant manifolds in these Hamiltonian PDEs, I will study two type of solutions: transfer of energy solutions, namely solutions that push energy to arbitrarily high modes as time evolves by drifting along resonances; and breathers, spatially localized and periodic in time solutions, which in a proper setting can be seen as homoclinic orbits to a stationary solution. Fields of science natural sciencesphysical sciencesastronomyplanetary sciencescelestial mechanicsnatural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equationsnatural sciencesmathematicsapplied mathematics Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-2017-STG - ERC Starting Grant Call for proposal ERC-2017-STG See other projects for this call Funding Scheme ERC-STG - Starting Grant Coordinator UNIVERSITAT DE BARCELONA Net EU contribution € 227 152,32 Address Gran via de les corts catalanes 585 08007 Barcelona Spain See on map Region Este Cataluña Barcelona Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Other funding € 0,00 Beneficiaries (2) Sort alphabetically Sort by Net EU contribution Expand all Collapse all UNIVERSITAT DE BARCELONA Spain Net EU contribution € 227 152,32 Address Gran via de les corts catalanes 585 08007 Barcelona See on map Region Este Cataluña Barcelona Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Other funding € 0,00 UNIVERSITAT POLITECNICA DE CATALUNYA Participation ended Spain Net EU contribution € 873 195,18 Address Calle jordi girona 31 08034 Barcelona See on map Region Este Cataluña Barcelona Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Other funding € 0,00
