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

研究人员引入了新的数学上严格的实验来计算混沌系统的动力学实体，并考虑了常用算法的严格可靠性。特别地，他们构造了适当的算法来计算混沌常微分方程组的轨道，如Lorenz方程，并在这种数值计算的轨道上推导出跟踪误差的严格估计；确定了由一步法得到的离散化系统的吸引子很好地逼近微分方程的吸引子的一般条件；并设计了一种严格的数值算法来测试特定动力系统中的双曲性。他们还研究同宿现象，这些现象在分析混沌行为中起着至关重要的作用。更具体地说，他们估计了单参数族耗散平面微分同态切的稠密子集上参数的区间，并证明了慢变系统中横截同宿点的存在性。动力系统中一些更值得注意的发现很大程度上是基于数值实验。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，人们已经意识到，自然界的许多行为都是混乱的。例如，某一天天气参数的微小变化，如温度和大气压，可能会导致几天后天气模式的大幅波动。描述自然界中的混沌系统的数学模型几乎不可避免地也会表现出混沌。由于这些数学模型很少能被精确求解，因此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，以确定计算机模拟的混沌系统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.