Frontiers in Dynamical Systems

动力系统是现代数学的一个分支，研究自然过程和迭代过程的时间演化。它既是纯数学的一个领域，也是多个科学学科交叉的一个学科。作为纯数学的一个领域，它旨在发展适用于一般过程的严格理论和技术，解决稳定性和混沌的基本问题。作为科学的合作伙伴，它提供了定性的理论和观点，当与数值计算结合使用时，它可以成为一个强大的工具。虽然动力系统的这两个方面本身都有价值，但理论和应用之间往往存在差距。拟议的研究既有纯理论和面向应用的组件，其目标之一是思想的交叉施肥。这项研究将：（1）导致动力系统理论的范围显着扩大，（2）加强动力系统和其他数学领域之间的联系，如概率和偏微分方程，（3）建立数学与统计物理和理论神经科学之间的联系。在这个建议中有四个具体的研究项目。有人建议将双曲理论（或混沌动力系统理论）从有限维扩展到无限维，这样扩展后的理论不仅适用于常微分方程，而且适用于大类偏微分方程。第二个项目建议扩展目前的技术，用于分析混沌动力学，这往往表现出高斯型波动，系统产生重尾统计。虽然公认为复杂过程的一个标志，这种动态过程的分析研究很少进行。第三个项目试图为非平衡统计力学中的某些基本现象学提供严格的理由，例如稳态动力学中的局部热平衡思想。第四个也是最后一个项目提出了在以前的神经科学计算模型，特别是在视觉皮层模型中识别的现象的数值和分析研究。为青年研究人员提供科学培训是拟议活动的一个非常重要的部分。首席研究员有一个强大的和有据可查的指导学生和初级同事的历史，她完全希望继续这样做，使用项目在目前的赠款。

Dynamical systems is a branch of modern mathematics concerned with time evolutions of natural and iterative processes. It is both an area of pure mathematics and a subject that lies at the crossroads of multiple scientific disciplines. As an area of pure mathematics, it seeks to develop rigorous theories and techniques that are applicable to general processes, addressing foundational issues of stability and chaos. As a partner with the sciences, it offers qualitative theories and viewpoints, and can be a powerful tool when used in conjunction with numerical computations. While both aspects of dynamical systems are worthy in their own right, there often is a gap that lies between theory and applications. The proposed research has both purely theoretical and application oriented components, one of its goals being the cross-fertilization of ideas. This research will: (1) lead to significant broadening of the scope of dynamical systems theory, (2) strengthen connections between dynamical systems and other areas of mathematics, such as probability and partial differential equations, and (3) build connections between mathematics and statistical physics and theoretical neuroscience. In this proposal there are four specific research projects are presented. One proposes to extend hyperbolic theory (or the theory of chaotic dynamical systems) from finite to infinite dimensions, so that an enlarged theory will be applicable not just to ordinary differential equations but also to large classes of partial differential equations. A second project proposes to extend current techniques for analyzing chaotic dynamics, which often exhibit Gaussian type fluctuations, to systems that produce heavy-tailed statistics. Though well recognized as a hallmark of complex processes, few analytical studies of such dynamical processes have been carried out. A third project seeks to provide rigorous justification for certain fundamental phenomenology in nonequilibrium statistical mechanics, such as the idea of local thermal equilibrium in steady states dynamics. The fourth and final project proposes numerical and analytical studies of phenomena identified in previous computational models of neuroscience, specifically in models of visual cortex. Providing scientific training for young researchers is a very important part of the proposed activity. The principal investigator has a strong and well documented history of mentoring students and junior colleagues, and she fully expects to continue to do so using projects in the current grant.