Working Paper by Mark Williamson and Tim Lenton
We generalize a method of detecting an approaching bifurcation in a timeseries of a noisy system from the one dynamical variable to the multi dynamical variable case. For a system described by one dynamical variable, the perturbations away from the system’s fixed point caused by noise will decay slower the closer the system is to a bifurcation. This phenomenon is known as critical slowing down and all systems described by a first order differential equation (ODE) with one dynamical variable can exhibit this decay-type behaviour.
However, when one has multiple coupled dynamical variables or equivalently second or higher order ODEs,the possible dynamics can be much richer, exhibiting oscillatory and chaotic behaviour. In our generalization to the multi-variable case we find additional indicators to decay rate, such as frequency of oscillation.
In the case of approaching a homoclinic bifurcation, there is no change in decay rate but there is a decrease in frequency of oscillations. The expanded method therefore adds extra tools to help detect and classify approaching bifurcations given one or more timeseries, where the underlying dynamics are not fully known.
The method is applicable to any set of timeseries regardless of its origin but may be particularly useful when anticipating abrupt changes in the multi-dimensional climate system.