Bifurcation Theory and Abrupt Climate Change

在数学方程系统的研究中，分岔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。分岔理论告诉我们，这种“因果”关系可以是高度非线性的，但通常只会以几种一般的方式发生，称为分岔，数学家们开始很好地理解这一点。在本工作中，所研究的方程是非线性微分方程组；也就是说，决定系统如何随时间演化的方程。这类方程在科学中被广泛应用，并有助于理解许多种类的实际系统，例如物理、工程、生物和气候。这个研究项目有两个目标。在数学领域，它将有助于分岔理论本身的进步，通过向知识库中添加新的分岔类型。在数学之外，这个研究项目将把分岔理论的思想应用到现实世界的系统中，目的是更好地理解这些系统是如何变化的。从这个角度研究的系统的一些例子是：全球气候变化，机械系统的振动模式，人类某些心肺疾病的发病，以及生态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计划的第一优先事项。最近，我和同事们已经确定，地球的气候系统可能会出现分叉，而且在遥远的过去确实发生过。一项初步分析得出的结论是，除非由二氧化碳排放增加引起的人为强迫得到严格限制，否则未来我们将面临急剧变暖的气候状态。这一结果为政府间气候变化专门委员会发布的可怕预测增加了数学上的严谨性和严肃性。建议使用更复杂的气候模型层次进一步调查这一灾难性的分岔及其缓解措施。

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.