Probing rare physical trajectories with Lyapunov weighted dynamics

In nonlinear dynamical systems, atypical trajectories often play an important role. For instance, resonances and separatrices determine the fate of planetary systems, and localized objects such as solitons and breathers provide mechanisms of energy transport in systems such as Bose - Einstein condensates and biological molecules. Unfortunately, most of the numerical methods to locate these 'rare' trajectories are confined to low-dimensional or toy models, whereas the realms of statistical physics, chemical reactions or astronomy are still hard to reach. Here we implement an efficient method that enables us to work in higher dimensions by selecting trajectories with unusual chaoticity. As an example, we study the Fermi - Pasta - Ulam nonlinear chain in equilibrium and show that the algorithm rapidly singles out the soliton solutions when searching for trajectories with low levels of chaoticity, and chaotic breathers in the opposite situation. We expect the scheme to have natural applications in celestial mechanics and turbulence, where it can readily be combined with existing numerical methods.