Importance sampling with imperfect cloning for the computation of generalized Lyapunov exponents

We revisit the numerical calculation of generalized Lyapunov exponents, L(q), in deterministic dynamical systems. The standard method consists of adding noise to the dynamics in order to use importance sampling algorithms. Then L(q) is obtained by taking the limit noise-amplitude → 0 after the calculation. We focus on a particular method that involves periodic cloning and pruning of a set of trajectories. However, instead of considering a noisy dynamics, we implement an imperfect (noisy) cloning. This alternative method is compared with the standard one and, when possible, with analytical results. As a workbench we use the asymmetric tent map, the standard map, and a system of coupled symplectic maps. The general conclusion of this study is that the imperfect-cloning method performs as well as the standard one, with the advantage of preserving the deterministic dynamics.