This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 757802.

The ERC’s mission is to encourage the highest quality research in Europe through competitive funding and to support investigator-driven frontier research across all fields, on the basis of scientific excellence.

The ERC complements other funding activities in Europe such as those of the national research funding agencies, and is a flagship component of Horizon 2020, the European Union’s Research Framework Programme for 2014 to 2020.

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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 a framework 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 analysed the role of resonances in this model. Nevertheless, mathematical results showing instabilities along them are rather scarce. This ERC project plans to develop a new theory to analyse the transversal intersection between invariant manifolds along mean motion and secular resonances to prove the existence of Arnold diffusion. This theory will also be applied to construct oscillatory motions.

Several Partial Differential Equations such as the nonlinear Schrodinger, 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 HamiltonianPDEs, two type of solutions will be studied: 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.

Host Institution

The Universitat Politècnica de Catalunya · BarcelonaTech (UPC) is the technical university of Catalonia with its base campus located in the city of Barcelona. It is one of the largest technical universities of southern EU, with over 30,000 students spanning all levels from undergraduate to PhD, and employing over 2,500 faculty. Its leadership in research and innovation projects, the rigour of its researchers and its network of cutting-edge facilities and equipment are the basis of the UPC’s positions in the main international scientific output rankings.

The Mathematics Department of UPC is one of its largest departments, and hosts strong research groups in dynamical systems and Partial Differential Equations, among others. The research group in dynamical systems is involved in the activities of the Barcelona Graduate School of Mathematics (BGSMath), which is a platform that coordinates the mathematical academic activities and PhD programmes of the three public universities within the area of Barcelona that have mathematics departments (UB, UAB and UPC).

Marcel Guardia

Principal Investigator

Since 2017, Marcel Guardia is Associate Professor at Universitat Politècnica de Catalunya (UPC), where he is a member of the Dynamical Systems group at UPC and the Lab of Geometry and Dynamical Systems. Dr. Guardia is also member of the BGSMath.

Before being at UPC, he held post-doctoral positions at Penn State University, Fields Insitute (University of Toronto), University of Maryland at College Park, Institute for Advanced Study (Princeton) and Université de Paris 7 – Diderot.

His research interests are Hamiltonian Systems, Celestial Mechanics, Hamiltonian Partial Differential Equations, Arnol’d diffusion, Exponentially small phenomena.

Further information on Dr. Guardia previous publications and research career, please refer to his personal website.