Dear Riccardo
Welcome on UQworld!
UQ techniques are in general general-purpose. Monte Carlo simulation is the simplest approach and it can help you start if your model of the dynamical system is fast to run.
Very often though, each single run may take minutes to hours, and you then need surrogate models. That’s were it becomes (sometimes) tricky with dynamical systems, since the standard techniques (e.g. polynomial chaos expansions, Gaussian processes, support vector machines) may not work out of the box to surrogate a full response trajectory.
I can suggest some of our recent papers in this field (with plenty of references inside), see below. In any case, feel free to ask further questions on this forum! All these analysis (from Monte Carlo to surrogate modelling to sensitivity analysis, etc.) can be carried out with UQLab.
Best regards
Bruno
[1] Mai, C. & Sudret, B., Surrogate models for oscillatory systems using sparse polynomial chaos expansions and stochastic time warping, SIAM/ASA J. Unc. Quant., 2017, 5, 540-571.
[2] Mai, C.-V., Spiridonakos, M. D., Chatzi, E. & Sudret, B., Surrogate modeling for stochastic dynamical systems by combining nonlinear autoregressive with exogeneous input models and polynomial chaos expansions, Int. J. Uncertainty Quantification, 2016, 6, 313-339.
[3] Yaghoubi, V., Marelli, S., Sudret, B. & Abrahamsson, T., Sparse polynomial chaos expansions of frequency response functions using stochastic frequency transformation, Prob. Eng. Mech., 2017, 48, 39-58.