Natural measures of asymptotically autonomous systems

Rate-induced phenomena can be mathematically modelled in terms of a dynamical system with a real-time (as opposed to quasistatic) parameter drift between two values; that is to say, the parameter converges to two different values as time tends to negative and positive infinity, giving rise to a nonautonomous dynamical system that is asymptotically autonomous. Representing stable climate states by attractors of the parameter-dependent autonomous system through which the parameter drift takes place, rate-induced tipping is modelled as the phenomenon that a trajectory of the nonautonomous system that starts in the past in the vicinity of one attractor lands in the vicinity of an attractor representing a different stable climate state in the future. However, if these attractors are chaotic, they exhibit sensitive dependence on initial conditions, which on the one hand makes investigation of any individually selected typical initial condition numerically impossible and physically irrelevant, but on the other hand makes a probabilistic description of long-term behaviour of trajectories an effective tool. This probabilistic description is provided by the "natural measure" on a chaotic attractor; in this poster, we consider the question of when this concept of "natural measures" can be extended from the classical setting of autonomous systems to the setting of asymptotically autonomous systems and hence used to provide a mathematically well-defined quantification of the "probability of tipping" between two stable climate states.