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

Christian Kuehn; Giuseppe Malavolta; Martin Rasmussen

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arXiv · 1803.00382

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

Christian Kuehn·Giuseppe Malavolta·Martin Rasmussen

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Abstract

We consider discrete-time one-dimensional random dynamical systems with bounded noise, which generate an associated set-valued dynamical system. We provide necessary and sufficient conditions for a discontinuous bifurcation of a minimal invariant set of the set-valued dynamical system in terms of the derivatives of the so-called extremal maps. We propose an algorithm for reconstructing the derivatives of the extremal maps from a time series that is generated by iterations of the original random dynamical system. We demonstrate that the derivative reconstructed for different parameters can be used as an early-warning signal to detect an upcoming bifurcation, and apply the algorithm to the bifurcation analysis of the stochastic return map of the Koper model, which is a three-dimensional multiple time scale ordinary differential equation used as prototypical model for the formation of mixed-mode oscillation patterns. We apply our algorithm to data generated by this map to detect an upcoming transition.

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BibTeX RIS

Christian Kuehn, Giuseppe Malavolta, Martin Rasmussen. 2018-03-01. Early-warning signals for bifurcations in random dynamical systems with bounded noise. https://arxiv.org/abs/1803.00382

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