Bifurcation Theory and Abrupt Climate Change

在数学方程组的研究中，分叉理论处理系统参数的微小变化可能导致方程解的定性（即拓扑）变化的情况。 分叉理论告诉我们，这种“因果”关系可以是高度非线性的，但通常只会以几种常见的方式发生，称为分叉，数学家们开始非常好地理解。 在目前的工作中，所研究的方程是非线性微分方程系统，也就是说，方程，确定系统如何随时间演变。 这类方程在科学中有着广泛的应用，有助于理解许多真实的系统，例如物理学、工程学、生物学和气候学。 这项研究计划有两个目标。 在数学领域，它将有助于分支理论本身的进步，通过增加新类型的分支知识库。 在数学之外，该研究计划将把分叉理论的思想应用到现实世界的系统中，目的是更好地了解这些系统可能会如何变化。 从这个角度研究的系统的一些例子是：全球气候变化，机械系统中的振动模式，人类某些心肺疾病的发病，以及生态学中的可持续生物多样性。 在所有这些应用中，气候变化被选为本研究提案的第一优先事项。 最近，我和我的同事们已经确定，地球气候系统中可能会发生分叉，而且实际上在遥远的过去的气候中也发生过。 初步分析得出的结论是，除非由二氧化碳产量增加引起的人为强迫得到严重遏制，否则未来将出现一个显著变暖的气候状态。 这一结果为政府间气候变化专门委员会（Intergovernmental Panel on Climate Change）发布的可怕预测增添了数学上的严谨性和庄严性。 进一步调查这一灾难性的分歧和缓解建议，使用层次结构更复杂的气候模式。

In the study of systems of mathematical equations, Bifurcation Theory deals with cases in which small changes in the parameters of the system can lead to qualitative (i.e. topological) changes in the solutions of the equations. Bifurcation Theory tells us that this “cause and effect” relationship can be highly nonlinear, but typically will occur only in a few generic ways, called bifurcations, which mathematicians are beginning to understand very well. In the present work, the equations under study are systems of nonlinear differential equations; that is, equations that determine how a system evolves with time. Equations of this kind are widely used in science, and help in the understanding of many kinds of real systems, for example in physics, engineering, biology and climate. This research program has two objectives. In the field of mathematics, it will contribute to the advancement of Bifurcation Theory itself, by adding new types of bifurcations to the knowledge base. Outside of mathematics, this research program will extend the application of ideas of Bifurcation Theory to real-world systems, with the goal of better understanding how these systems are likely to change. Some examples of systems that have been investigated from this perspective are: global climate change, patterns of vibrations in mechanical systems, onset of certain cardio-respiratory diseases in humans, and sustainable biodiversity in ecology. Of all these applications, climate change is selected as the first priority for this research proposal. Very recently, coworkers and I have established that bifurcations may occur in the Earth's climate system, and actually have occurred in climates of the distant past. A preliminary analysis has concluded that a bifurcation to a dramatically warmer climate state awaits us in the future, unless anthropogenic forcing, caused by increased carbon dioxide production, is severely curtailed. This result adds mathematical rigour and gravitas to the dire predictions that have been issued by the Intergovernmental Panel on Climate Change. Further investigations of this catastrophic bifurcation and its mitigation are proposed, using a hierarchy of more sophisticated climate models.