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Rare events in non-linear dynamical systems are difficult to sample becauseof the sensitivity to perturbations of initial conditions and of complexlandscapes in phase space. Here we discuss strategies to control thesedifficulties and succeed in obtainining an efficient sampling within aMetropolis-Hastings Monte Carlo framework. After reviewing previous successesin the case of strongly chaotic systems, we discuss the case of weakly chaoticsystems. We show how different types of non-hyperbolicities limit theefficiency of previously designed sampling methods and we discuss strategieshow to account for them. We focus on paradigmatic low-dimensional chaoticsystems such as the logistic map, the Pomeau-Maneville map, and area-preservingmaps with mixed phase space.

Taming chaos to sample rare events: the effect of weak chaos