Mathematical Sciences: Computational Algorithms in Chaotic Dynamical Systems and Homoclinic Phenomena

研究人员引入了新的数学严谨的实验来计算混沌系统的动态实体，并考虑了常用算法的严格可靠性。特别是，他们构建了合适的算法来计算混沌常微分方程的轨道，如洛伦兹方程，并推导出在这种数值计算轨道上的阴影误差的严格估计；确定由一步法得到的离散系统的吸引子很好地逼近微分方程的吸引子的一般条件；并设计了一个严格的数值算法来测试特定动力系统的双曲性。他们也研究同斜现象，这在混沌行为的分析中起着重要的作用。更具体地说，他们估计了一个密度子集上的参数区间，其中一个单参数耗散平面微分同态族具有同斜切点，并建立了慢变系统中横向同斜点的存在。动力系统中一些比较值得注意的发现很大程度上是基于数值实验的。重要的是设计这样的实验，其可靠性可以得到严格的验证。这个项目的大部分内容都与混沌系统有关。近年来，人们已经认识到自然界的许多行为都是混沌的。例如，某一天天气参数（如温度和气压）的微小变化可能导致几天后天气模式的大幅波动。描述自然界混沌系统的数学模型几乎不可避免地也会表现为混沌。因为这些数学模型很少能被精确地解决，所以对它们行为的了解在很大程度上依赖于计算机模拟。这个项目的主要目标是开发测试，以告诉混沌系统的计算机模拟与系统的真实行为有多接近。

The investigators introduce new mathematically rigorous experiments to compute dynamical entities of chaotic systems, and they consider the rigorous reliability of commonly used algorithms. In particular, they construct suitable algorithms to compute orbits of chaotic ordinary differential equations, such as the Lorenz equations, and derive rigorous estimates of the shadowing error in such numerically computed orbits; determine general conditions under which an attractor of a differential equation is well-approximated by an attractor of the discretized system resulting from a one-step method; and devise a rigorous numerical algorithm to test for hyperbolicity in specific dynamical systems. They also study homoclinic phenomena, which play an essential role in the analysis of chaotic behavior. More specifically, they estimate the intervals of the parameter on a dense subset of which a one-parameter family of dissipative planar diffeomorphisms has homoclinic tangencies, and establish the presence of transversal homoclinic points in slowly varying systems. Some of the more noteworthy discoveries in dynamical systems have largely been based on numerical experiments. It is important to devise such experiments whose reliability can be rigorously verified. Most of this project is concerned with chaotic systems. In recent years it has been realized that much of nature behaves chaotically. For example, small changes on a given day in weather parameters such as temperature and barometric pressure can lead to large fluctuations in weather patterns a few days later. A mathematical model that describes a chaotic system in nature will almost inevitably behave chaotically also. Because these mathematical models can rarely be solved exactly, knowledge of their behavior relies greatly on computer simulations. The main object of this project is to develop tests that will tell how close a computer simulation of a chaotic system is to a true behavior of the system.