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

Title: 非自治混沌系统在非周期性强迫下的演化：一个气候变化的例子

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. Show Chinese abstract

Abstract: 复杂地球系统模型被广泛用于在某些关于未来大气温室气体浓度变化的假设下，对未来的气候做出条件性陈述；这些陈述通常被称为气候投影。 这些模型本身是高维非线性系统，通常以吸引子和低维非线性系统（如经典的Lorenz 63系统）来讨论它们的行为。 在非自治情况下，例如由于人为气候变化，相关对象有时被认为是拉回吸引子或快照吸引子。 然而，拉回吸引子是在给定时间的一组{\em 全部}可能的系统状态的集合，因此在进行气候投影时没有考虑我们对地球系统当前状态的知识，因此对于年度到数十年的气候投影来说并不十分有用。 在本文中，我们通过使用一个具有三个时间尺度变化和非周期性强迫的低维、类似气候的非线性系统，来研究测量和解释模型中期气候的问题。 我们引入了一个{\em 进化集}的概念，该概念依赖于系统的起始状态，并探讨了它与不同类型初始条件不确定性和外部强迫速率之间的联系。 我们将{\em 收敛时间}定义为其中一个因变量的分布失去其初始条件记忆所需的时间。 我们怀疑收敛时间和经典混合时间概念之间存在联系，但这种联系的精确性质需要进一步探索。 这些结果对设计有影响力的气候和地球系统模型集合具有意义，并引发了一些数学上感兴趣的问题。