title: Adapting to a changing environment: non-obvious thresholds in multi-scale systems.
abstract: Many natural and technological systems fail to adapt to changing external conditions and move to a different state if the conditions vary too fast. Such 'non-adiabatic' processes are ubiquitous, but little understood. We identify these processes with a new nonlinear phenomenon-an intricate threshold where a forced system fails to adiabatically follow a changing stable state. In systems with multiple time scales, we derive existence conditions that show such thresholds to be generic, but non-obvious, meaning they cannot be captured by traditional stability theory. Rather, the phenomenon can be analysed using concepts from modern singular perturbation theory: folded singularities and canard trajectories, including composite canards. Thus, non-obvious thresholds should explain the failure to adapt to a changing environment in a wide range of multi-scale systems including: tipping points in the climate system, regime shifts in ecosystems, excitability in nerve cells, adaptation failure in regulatory genes and adiabatic switching in technology.
authors.affiliation: Mathematics Research Institute, University of Exeter , Exeter EX4 4QF, UK.
authors.forename: Clare
authors.initials: C
authors.lastname: Perryman
authors.affiliation: Mathematics Research Institute, University of Exeter , Exeter EX4 4QF, UK ; Department of Applied Mathematics , Western Gateway Building, University College Cork , Ireland.
authors.forename: Sebastian
authors.initials: S
authors.lastname: Wieczorek
keywords: canards; folded singularity; rate-induced bifurcations; thresholds