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We consider discrete-time one-dimensional random dynamical systems withbounded noise, which generate an associated set-valued dynamical system. Weprovide necessary and sufficient conditions for a discontinuous bifurcation ofa minimal invariant set of the set-valued dynamical system in terms of thederivatives of the so-called extremal maps. We propose an algorithm forreconstructing the derivatives of the extremal maps from a time series that isgenerated by iterations of the original random dynamical system. We demonstratethat the derivative reconstructed for different parameters can be used as anearly-warning signal to detect an upcoming bifurcation, and apply the algorithmto the bifurcation analysis of the stochastic return map of the Koper model,which is a three-dimensional multiple time scale ordinary differential equationused as prototypical model for the formation of mixed-mode oscillationpatterns. We apply our algorithm to data generated by this map to detect anupcoming transition.

Early-warning signals for bifurcations in random dynamical systems with bounded noise