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Bifurcations of random dynamical systems with bounded noise

具有有限噪声的随机动力系统的分岔

基本信息

批准号：
EP/W009455/1
负责人：
Jeroen Lamb
金额：
$ 10.28万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW009455%2F1
关键词：
Bifurcations random dynamical systems bounded

项目摘要

The development of dynamical systems theory has been one of the scientific revolutions of the 20th century, and many insights from this field are now at the heart of deep and abstract mathematics as well as computational methodologies in many branches of science and engineering. During the last decades, the importance of considering the presence of noise and uncertainty has become evident in real-world applications, including the life sciences and artificial intelligence, but a corresponding random dynamical systems theory is still only in its very early stages of development.Bifurcation theory addresses the fundamental question of how the dynamics of a system changes under variation of parameters. While bifurcation theory is a cornerstone of the deterministic theory of dynamical systems, a corresponding theory in the presence of noise is still in its infancy, despite its relevance to many topical applications. This is due to the fact that the deterministic theory does not generalise in any obvious way to the random setting.The compound behaviour of a random dynamical system with bounded noise - describing the collection of trajectories with all possible noise realisations - admits a description at the topological level as a deterministic set-valued dynamical system. Attractors of this set-valued system correspond to attractors of the associated random system. However, set-valued dynamical systems are notoriously difficult to analyse, since they are defined on the set of all compact subsets, which is not a Banach space. This prevents the use of the implicit function theorem and other powerful tools fundamental for bifurcation theory (and, in general, qualitative dynamical systems theory) and is a well-known obstacle for theoretical and numerical methods alike. As a consequence, the tracking and bifurcation analysis of attractors of set-valued dynamical systems is a notoriously hard problem.We propose to use a novel insight to resolve the challenge of bifurcation analysis in set-valued dynamics, namely that boundaries of attractors of a dynamical system with bounded noise (and its associated set-valued dynamical system) admit a representation in terms of projected invariant manifolds of a related finite-dimensional dynamical system, defined on the unit tangent bundle over the state space. This boundary map opens up the possibility of studying bifurcations of attractors of systems with bounded noise, thus enabling the use of existing techniques of local and global bifurcation theory in the latter setting.

动力系统理论的发展是20世纪的科学革命之一，该领域的许多见解现在是科学和工程许多分支的深奥和抽象数学以及计算方法的核心。在过去的几十年里，考虑噪声和不确定性的重要性已经在现实世界的应用中变得明显，包括生命科学和人工智能，但相应的随机动力系统理论仍然只是在其发展的早期阶段。分岔理论解决了系统动力学在参数变化下如何变化的基本问题。虽然分岔理论是动力系统确定性理论的基石，但在噪声存在下的相应理论仍处于起步阶段，尽管它与许多局部应用相关。这是由于确定性理论不能以任何明显的方式推广到随机设置的事实。具有有界噪声的随机动力系统的复合行为-描述具有所有可能噪声实现的轨迹集合-允许在拓扑水平上描述为确定性集值动力系统。该集值系统的吸引子对应于相关随机系统的吸引子。然而，集值动力系统是出了名的难以分析，因为它们是在所有紧子集的集合上定义的，而不是巴拿赫空间。这阻碍了隐函数定理和其他强大的分岔理论（一般来说，定性动力系统理论）基础工具的使用，并且是理论和数值方法的一个众所周知的障碍。因此，集值动力系统吸引子的跟踪和分岔分析是一个非常困难的问题。我们建议使用一种新的见解来解决集值动力学中分岔分析的挑战，即具有有界噪声的动力系统（及其相关的集值动力系统）的吸引子边界允许用相关有限维动力系统的投影不变流形表示，该流形定义在状态空间上的单位切线束上。该边界图开辟了研究有界噪声系统吸引子分岔的可能性，从而使局部和全局分岔理论的现有技术能够在后者的设置中使用。