PANDA (Patterns, Nonlinear Dynamics and Applications)

Tuesday 3 June 2014

Department of Mathematical Sciences

University of Bath

Sebastian Wieczorek (Exeter)

Rate-induced bifurcations: critical rates, non-obvious thresholds, and adaptation failure

 
Abstract: Many natural and technological systems fail to adapt to changing external conditions and move to a new state if the conditions vary too fast. I will conceptualise the failure to adapt as a rate-induced bifurcation --- a nonautonomous instability characterised by critical rates of change and instability thresholds. Specifically, such instabilities occur in systems with a (unique) stable state that exists continuously for all fixed values of the external conditions. When external conditions vary in time, the position of the stable state changes and the forced system tries to keep pace with the changes. However, some systems fail to track the continuously changing stable state and destabilise above some critical rate of change.

Scientists often find rate-induced bifurcations counter-intuitive because there is no obvious loss of stability. Mathematically, rate-induced bifurcations cannot in general be described by classical bifurcation theory or asymptotic approaches. Thus, they require an alternative approach. I will present an approach based on geometrical singular perturbation theory to study critical rates of change and (non-obvious) thresholds in terms of connecting (heteroclinic) orbits and folded singularities. The mathematical approach will be illustrated using examples of the "critical rate hypothesis" in herbivore-plant interaction and the "compost-bomb instability" in climate-carbon cycle. I will also discuss repercussions for climate change policy making which currently focuses on critical levels of the atmospheric temperature whereas the critical factor may be the rate of warming rather than the temperature itself.