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

TL;DR

This paper develops a method to detect early-warning signals of bifurcations in one-dimensional random dynamical systems with bounded noise by reconstructing derivatives of extremal maps from time series data.

Contribution

It introduces necessary and sufficient conditions for bifurcations and proposes an algorithm to reconstruct extremal map derivatives as early-warning signals from data.

Findings

Reconstructed derivatives serve as early-warning signals for bifurcations.

Algorithm successfully detects upcoming bifurcations in the Koper model.

Method applicable to systems with bounded noise and set-valued dynamics.

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.