Set-valued numerical analysis for critical transitions

十多年来，复杂系统研究一直是许多国家和国际研究委员会的科学优先事项。许多非常有趣的现象已经被发现和探索，但基础数学理论的发展一直滞后。拟议的研究建立在一个新兴的运动，应用数学，其目的是提供适当的数学分岔理论的存在的预警信号的突然变化的动力学行为。这些突然的变化通常被称为临界转变，并已被应用科学家在各种情况下报道。这种预警信号的存在具有深远的实际意义，因为这将使人们能够制定更好的控制战略，以避免或减少灾难的影响，在本项目中，将发展临界转变的数值研究技术。特别是，它将提供新的技术近似不变集。这项研究将是基于最近的结果，使代表性的不变集的功能，在Banach空间可能。这种新方法的主要优点之一是，它可以用来研究分叉，相比之下，网格单元离散化，这是目前最先进的不变对象的计算。有界噪声随机系统的具体数值研究将导致洞察如何早期预警可以给出一个动态系统的方法分叉点。

Complex systems research has been on the forefront of scientific priorities of many national and international research councils for more than a decade. Many very interesting phenomena havebeen identified and explored, but the development of the underpinning mathematical theory has been lagging behind. The proposed research builds on an emerging movement in appliedmathematics which aims to provide proper mathematical bifurcation theory for the existence of early-warning signals for sudden changes in dynamical behaviour. These sudden changes are commonlyreferred to as critical transitions, and have been reported by applied scientists in various contexts. Practical implications for the existence of such early-warning signals are far reaching, since these would enable the development of better control strategies to avoid or diminish the effect of catastrophes.In this project, techniques for the numerical study of critical transitions will be developed. In particular, it will provide new techniques for the approximation of invariant sets. The research will be based on very recent results that make a representation of such invariant sets as functions in a Banach space possible. One of the main advantages of this new approach is that it can be used to study bifurcations, in contrast to grid-cell discretisations, which is the current state-of-the-art for the computation of invariant objects. The specific numerical studies on random systems with bounded noise will lead to insights into how an early-warning can be given should a dynamical system approach a bifurcation point.