Earlywarning signals for bifurcations in random dynamical systems with bounded noise
Abstract
We consider discretetime onedimensional random dynamical systems with bounded noise, which generate an associated setvalued dynamical system. We provide necessary and sufficient conditions for a discontinuous bifurcation of a minimal invariant set of the setvalued dynamical system in terms of the derivatives of the socalled 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 earlywarning 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 threedimensional multiple time scale ordinary differential equation used as prototypical model for the formation of mixedmode oscillation patterns. We apply our algorithm to data generated by this map to detect an upcoming transition.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1803.00382
 Bibcode:
 2018arXiv180300382K
 Keywords:

 Mathematics  Dynamical Systems;
 37G35;
 37H20;
 37C70;
 49K21;
 70K70