Bifurcations of random dynamical systems with bounded noise

动力系统理论的发展已经成为20世纪世纪的科学革命之一，并且来自该领域的许多见解现在处于科学和工程的许多分支中的深刻和抽象数学以及计算方法的核心。在过去的几十年中，考虑噪声和不确定性的重要性在现实世界的应用中已经变得明显，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处于发展的早期阶段。分岔理论解决了系统的动力学在参数变化下如何变化的基本问题。虽然分叉理论是动力系统的确定性理论的基石，但存在噪声的相应理论仍处于起步阶段，尽管其与许多热门应用相关。这是因为确定性理论并没有以任何明显的方式推广到随机环境中。随机动力系统与有界噪声的复合行为-描述所有可能的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。这个集值系统的吸引子对应于相关联的随机系统的吸引子。然而，集值动力系统是出了名的难以分析，因为它们被定义在所有紧子集的集合上，而不是Banach空间。这就阻碍了隐函数定理和其他强大的分支理论（以及一般的定性动力系统理论）的基本工具的使用，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支分析是一个非常困难的问题，我们提出了一个新的见解来解决集值动力系统分支分析的挑战，即有界噪声动力系统吸引子的边界（及其相关的集值动力系统）允许相关的有限维动力系统的投影不变流形的表示，定义在状态空间上的单位切丛上。这个边界映射开辟了研究有界噪声系统的吸引子的分叉的可能性，从而使使用现有的技术的局部和全局分叉理论在后者的设置。

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.