The evolving butterfly: Statistics in a changing attractor

Abstract

The Earth system is often modeled as a dynamical system in what has come to be known as Earth System Models. When used to study anthropogenically forced climate change, these models are forced in such a way that they are not in a statistically stationary state. Yet, statistical statements are still made about the Earth climate system using only a single trajectory by taking temporal averages. At each moment in time, one draws a sample from a different distribution, raising questions about the utility of temporal averages, in stark contrast to the utility of temporal averages in ergodic systems. This work follows in the tradition of using a toy model to examine properties present in the Earth climate system. We aim to examine how we can make meaningful statistical statements in non-stationary systems when only dealing with a single trajectory. We use the Lorenz equations with a time-varying parameter as a starting point for comparing ensemble averages to temporal averages. We find that, in so far as the control parameter induces a slow and smooth change in the dynamics, the resulting statistics of ensemble averages compare well to those of temporal averages.