I am a physical mathematician working in dynamical systems and equations of Physics whose interest is in detecting existence and conservations of periodic solutions and tori, and instability mechanisms coming from their destruction (chaos,(wave) turbulence, transfer of energy).In particular, I am involved in the matter of (in)stability with respect to perturbation of initial data for Anti-de Sitter spacetime (AdS).AdS is known to be not a physical spacetime due to lack of global hyperbolicity, namely it is impossible to forecast every event from some fixed region of this universe and moreover it is allowed to go back in time. However, because of the so called AdS-CFT Correspondence (Maldacena, 1998), knowledge of AdS dynamics has potential applications in condensed matter physics and elementary particles.From the papers of Bizon and Rostworowski (2011), and of Maliborski and Rostworowski (2013), it seems that on one hand there exist periodic solutions close to AdS forming therefore stability islands, on the other hand that under generic perturbation of intial data formation of black holes sets in and it seems that some solutions which are initially close to AdS and collapse to black hole perform a (wave) turbulent behaviour. As far as the latter is concerned, it happens that some solutions consist in superposition of nonlinearly interacting waves which are out-of-statistical equilibrium but such that their energy spectrum has a power-law decay in average: this has been observed in a number of physical phenomena, from hydrodynamics to plasma equations for instance, and it has a wide number of important applications, like in weather forecasts.My task is to provide a rigorous mathematical proof of these two things, supported also by correct numerics if necessary.