| The dynamical systems and differential equations that arise in theoretical physics often have a variational nature, for instance because nature seeks to minimize energy. We know that this variational structure often guarantees that solutions with prescribed geometric properties exist. Such particular solutions can include equilibrium points, limit cycles, connecting orbits and periodic patterns, that can be ``miminizers'' or ``saddle points''. Nevertheless, such simple solutions are relatively rare. In fact, we know from perturbation theory that ``quasi-periodic'' solutions are often more ample. The aim of this proposal is to develop a rigorous variational calculus for these quasi-periodic solutions and their generalizations. In particular, we would like to investigate the existence of saddle-type quasi-periodic solutions. Applications arise in the theory of parabolic recurrence relations, twist maps, solid state physics, Hamiltonian dynamics and reaction-diffusion equations. |
