Applied Nonautonomous Dynamical Systems: Theory, Methods and Examples

在气候建模方面，国际上已达成共识，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(气候公约)的目标是“将大气中的温室气体浓度稳定在可防止对气候系统造成危险的人为干扰的水平”，并在“足以使生态系统自然适应气候变化的时间框架内”做到这一点……政策讨论通常集中在这一目标的第一部分，这是关于定义必须避免的全球变暖的“危险”水平--最近的一次是通过2015年巴黎协议，该协议旨在将全球平均气温上升控制在远低于2摄氏度的水平。这项提议旨在制定方法，解决这一目标的第二部分，即了解系统的变化如何与系统的反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，由于正反照率反馈而失控的冰流失，或海洋热传输模式的重大变化。缓慢变化的强迫可能已经导致气候状态的相对快速变化，过去曾有过强迫参数(例如大气温室气体浓度)超过阈值的情况被认为是临界点。这些都是关于非自治动力系统的问题，即随时间演变的动力系统，但其中的参数也随时间变化。尽管近年来人们一致试图获得非自治动力系统的理论基础，但这些结果在具体应用中可能很难应用或不是特别有用：这是因为这一领域的大多数结果要么适用于非常一般的系统，要么适用于非常特定的系统。在与系统内的时间尺度类似的时间尺度上对系统进行更改(强制/输入)的情况下，特别少有适用的工具。这项提议旨在通过开发这样的工具来纠正这个问题。我们打算开发新的方法，以气候模拟中的应用为指导，根据Milnor类型的局部拉回吸引子及其对强迫(可能是非平稳的)的不稳定性来理解“典型行为”。我们将关注的特别问题是全球平均温度和温室气体对海洋环流的响应，特别是在强迫时间尺度与海洋环流时间尺度重合的情况下。

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.