This work aims to use tools of Dynamical Systems to prove that the nonlinear Schr odinger equation presents a global instability. To do that, we rst reduce this PDE to a system of ODE's of dimension N which we call the Toy Model System. Consequently our new purpose is to study the existence of instability in a system of ODE's. The way of proving it will consist in taking invariant objects and showing that there exists a solution that ows near all of them. This strategy resembles Arnold Di usion. The contribution of this work is to use the so-called Silnikov coordinates to prove this result when N = 3. However, we detect a problem when we ow around one of this invariant objects (that, in suitable coordinates, can be seen as a saddle) that prevents us from completing the proof in the expected way.

L'objectiu d'aquest treball és demostrar una inestabilitat que presenta l'equació de Schrödinger no lineal, convenientment reduïda a un sistema finit d'EDO's, mitjançant tècniques pròpies dels Sistemes Dinàmics, com seria l'ús de les anomenades coordenades de Shilnikov.. Es tracta d'utilitzar tècniques avanzades de sistemes dinàmics, concretamet el mètode de Shilnikov, per tal de trobar, de manera constructiva, solucions de l'equació de Schrodinger nolineal "cubic defocusing" amb transferència considerable d'energia i, en particular, aplicar-ho al model de l'article: J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Invent. Math., 181(1):39¿113, 2010.