Title: The evolution of a non-autonomous chaotic system under non-periodic forcing: a climate change example

Abstract: Complex Earth System Models are widely utilised to make conditional statements about the future climate under some assumptions about changes in future atmospheric greenhouse gas concentrations; these statements are often referred to as climate projections. The models themselves are high-dimensional nonlinear systems and it is common to discuss their behaviour in terms of attractors and low-dimensional nonlinear systems such as the canonical Lorenz `63 system. In a non-autonomous situation, for instance due to anthropogenic climate change, the relevant object is sometimes considered to be the pullback or snapshot attractor. The pullback attractor, however, is a collection of {\em all} plausible states of the system at a given time and therefore does not take into consideration our knowledge of the current state of the Earth System when making climate projections, and are therefore not very informative regarding annual to multi-decadal climate projections. In this article, we approach the problem of measuring and interpreting the mid-term climate of a model by using a low-dimensional, climate-like, nonlinear system with three timescales of variability, and non-periodic forcing. We introduce the concept of an {\em evolution set} which is dependent on the starting state of the system, and explore its links to different types of initial condition uncertainty and the rate of external forcing. We define the {\em convergence time} as the time that it takes for the distribution of one of the dependent variables to lose memory of its initial conditions. We suspect a connection between convergence times and the classical concept of mixing times but the precise nature of this connection needs to be explored. These results have implications for the design of influential climate and Earth System Model ensembles, and raise a number of issues of mathematical interest.