Applied Nonautonomous Dynamical Systems: Theory, Methods and Examples

在气候模拟方面，国际上一致认为有必要采取行动，减轻人为强迫造成的气候变化的影响。《联合国气候变化框架公约》（UNFCCC）的目标是“将大气中的温室气体浓度稳定在一个水平上，以防止气候系统受到危险的人为干扰”，并“在一个足够的时间框架内，使生态系统能够自然地适应气候变化....”。政策讨论通常集中在这一目标的第一部分，即定义必须避免的全球变暖的“危险”水平——最近通过2015年巴黎协议，该协议旨在将全球平均气温的上升幅度控制在远低于2摄氏度的水平。本建议旨在开发解决该目标第二部分的方法，即了解系统的变化如何与系统的响应时间框架相互作用。过去气候的突然变化（如古气候记录所证明的）和各组成部分之间的非线性反馈的存在，突出了气候系统的某些部分可能从一种状态“倾斜”到另一种状态的可能性——例如，由于正反照率反馈导致的失控的冰损失，或海洋热输送模式的重大变化。缓慢变化的强迫可能已经导致气候状态的相对快速变化，并且在过去，超过强迫参数的阈值（例如大气温室气体浓度）已与临界点有关。这些是关于非自治动力系统的问题，即动力系统随时间演化，但参数也随时间变化。尽管近年来已经有一个协调一致的尝试来获得非自治动力系统的理论基础，但这些结果可能很难应用或在特定应用中不是特别有用：这是因为该领域的大多数结果要么是非常一般的，要么是非常具体的系统。对于系统变更（强制/输入）发生的时间尺度与系统内的时间尺度相似的情况，很少有适用的工具。本提案旨在通过开发此类工具来纠正这一问题。我们打算开发新的方法，在气候模拟应用的指导下，根据米尔诺型的局部回拉吸引子及其对强迫的不稳定性（可能是非平稳的）来理解“典型行为”。我们将重点关注的具体问题是温室气体对全球平均温度和海洋环流对人为强迫的响应，特别是在强迫时间尺度与海洋环流时间尺度一致的情况下。

In climate modelling, there is international consensus on the need for action to mitigate the effects of climate change due to anthropogenic forcing. The United Nations Framework Convention on Climate Change (UNFCCC) has the objective to "stabilize greenhouse gas concentrations in the atmosphere at a level that would prevent dangerous anthropogenic interference with the climate system", and to do so "within a time-frame sufficient to allow ecosystems to adapt naturally to climate change....". Policy discussions generally focus on the first part of this objective, which is about defining a "dangerous" level of global warming which must be avoided - most recently through the 2015 Paris agreement which aims to hold the increase in the global average temperature to well below 2C. This proposal aims to develop methodologies that address the second part of this objective, namely to understand how changes to the system may interact with time-frame of response for the system.The presence of sudden changes in past climate (as evidenced in paleoclimate records) and nonlinear feedbacks between components has highlighted the likelihood that parts of the climate systems may "tip" from one state to another - for example, runaway ice-loss due to the positive albedo feedback, or major changes in ocean heat transport patterns. Slowly changing forcing may already lead to relatively fast changes in climate state, and passing through threshold value of a forcing parameter (for instance, in atmospheric greenhouse gas concentration) has been implicated in tipping points in the past. These are questions about nonautonomous dynamical systems, i.e. dynamical systems that evolve in time but where parameters also change with time. Although there has been a concerted attempt to gain a theoretical underpinning of nonautonomous dynamical systems in recent years, these results can be hard to apply or not particularly useful in specific applications: this is because most results in this area are either for very general, or for very specific systems. There are especially few applicable tools for cases where a change to the system (forcing/input) occurs on timescale similar to those within the system. This proposal aims to rectify this problem by developing such tools. We intend to develop new methods to understand "typical behaviour" in terms of local pullback attractors of Milnor type and their instabilities to forcing (that may be non-stationary), guided by applications in climate modelling. Particular problems we will focus on are the response of global mean temperature and ocean circulation by greenhouse gases to anthropogenic forcing, especially where the forcing timescale coincides with ocean circulation timescales.