Mathematical Sciences: Numerics for Dynamics

研究常微分方程的大多数数值方法是基于以下两种思想之一：(1)通过微分方程的近似来跟踪单个轨迹，或(2)通过求解相关的边值问题来搜索具有预定性质的轨道，例如周期轨道或不变环面。虽然这些技术显然是成功的，但它们也有缺点；人们只能在有限的时间间隔内近似轨道，而且这些方法很难检测出在动力学意义上不稳定的轨道。这些不稳定的轨道可能非常重要，因为它们定义了引力盆地的边界，从而决定了系统的整体动力学。此外，在许多问题中，感兴趣的对象恰好是不稳定的有界轨道，例如行波。在第二种情况下，我们需要先验地知道应该寻找哪种类型的解以及应该在相空间的哪个区域寻找解。研究者开发了研究常微分方程整体动力学的补充方法。该项目的理论基础是康利的动力系统理论。这项工作分为四个任务。(1)给定一个常微分方程和相空间中的一个区域作为输入，开发有效的数值格式来确定隔离邻域并计算其康利指数。(2)给定一个潜在的隔离邻域，开发更有效的机制来获得计算机辅助证明，以验证该邻域是一个隔离邻域，并严格计算相关隔离不变量集的Conley指数。(3)推广了抽象的Conley指数理论，使人们能够从指数和孤立邻域的知识中得出关于动力学结构的结论。(4)将这些技术应用于对全局动力学知之甚少的常微分方程。应该强调的是，这四个步骤的最终目标是对常微分方程的全局动力学有一个大致的了解。特别是，这些程序提供了关于发生的动力学的类型和结构以及这些动力学发生的相空间区域的信息。然后，可以使用开始时提到的更经典的方法更有效地研究对这些动力学的更好理解。很明显，大多数现象，无论是物理的、化学的还是生物的，都需要用非线性模型来描述。因此，要描述这些现象的动态，就需要解决全局分析中的问题。在实践中，这通常需要数值计算，因为我们的理论分析技术还不够充分。随着高性能计算的出现，在理解各种非线性动力系统的行为方面取得了相当大的进展。然而，目前我们所掌握的大多数数值和计算技术都有许多缺点。这个项目的目标是其中的两个。首先，经典的数值技术往往在很长的时间间隔内提供不好的近似（即，它们对于长期预测是不可靠的），而且它们难以检测不稳定的动态结构（这些不稳定的结构决定了被建模对象达到其渐近状态的路径）。第二个问题涉及难以理解计算机能够产生的大量数值数据。通常这个问题可以通过图形化显示数据来解决。不幸的是，大多数模型涉及三个以上的变量，因此可视化数值结果成为一项困难的任务。本项目发展了描述由非线性常微分方程引起的全局动力学的理论技术和有效的数值方法，然后将这些技术应用于广泛的生物和物理问题。理论方面涉及将代数信息转化为动力学，而数值方面涉及通过近似微分方程来计算这些代数不变量。代数信息同样适用于稳定或不稳定的动力学，当然，作为代数，不需要为了理解而可视化。

Mischaikow Most numerical methods for studying ordinary differential equations are based on one of the following two ideas: (1) tracking individual trajectories via an approximation of the differential equation, or (2) searching for orbits with predetermined properties, e.g. a periodic orbit or an invariant torus, by solving an associated boundary value problem. While these techniques are obviously successful, they have shortcomings; one can only approximate orbits over finite time intervals and these methods have difficulty detecting orbits which are unstable in the sense of dynamics. These unstable orbits can be extremely important since they define the boundaries of basins of attraction, and hence, determine the global dynamics of the system. In addition, there are many problems where the objects of interest are precisely the unstable bounded orbits, e.g. traveling waves. In the second case, one needs to know a priori what types of solutions should be looked for and in what regions of phase space one should look for the solutions. The investigator develops complimentary methods for studying the global dynamics of ordinary differential equations. The theoretical basis for the project is Conley's theory of dynamical systems. The work is separated into four tasks. (1) Given as an input an ordinary differential equation and a region in phase space, develop efficient numerical schemes to determine isolating neighborhoods and compute their Conley indices. (2) Given a potential isolating neighborhood, develop more efficient mechanisms for obtaining computer assisted proofs to verify that the neighborhood is an isolating neighborhood and to rigorously compute the Conley index of the associated isolated invariant set. (3) Extend the abstract Conley index theory, which allows one to draw conclusions concerning the structure of the dynamics from knowledge of the index and the isolating neighborhood. (4) Apply these techniques to ordinary differential equations for which the global dynamics is poorly understood. It should be emphasized that the end goal of these four steps is to obtain a rough understanding of the global dynamics of the ordinary differential equation. In particular, these procedures provide information about both the type and structure of dynamics that occur, and the region of phase space in which these dynamics take place. A finer understanding of these dynamics can then be more efficiently investigated using the more classical methods mentioned at the beginning. It is clear that most phenomena, whether physical, chemical or biological in nature, need to be described by nonlinear models. Thus, to describe the dynamics of these phenomena one is required to solve problems in global analysis. In practice, this usually requires numerical computations, since our theoretical analytic techniques are not yet adequate. With the advent of high performance computing considerable progress has been made in understanding the behavior of a wide variety of nonlinear dynamical systems. However, there are many drawbacks to most current numerical and computational techniques at our disposal. This project is aimed at two of them. The first arises from the fact that the classical numerical techniques often provide bad approximations over long time intervals (i.e. they are unreliable for long range predictions) and also they have difficulty detecting unstable dynamic structures (these unstable structures determine the path by which the object being modelled reaches its asymptotic state). The second problem involves difficulty in understanding the tremendous amounts of numerical data which the computers are capable of generating. Typically this problem is overcome by graphically displaying the data. Unfortunately, most models involve more than three variables, and hence visualizing the numerical results becomes a difficult task. This project develops theoretical techniques and efficient numerical methods for describing the global dynamics arising from nonlinear ordinary differential equations, and then applies these techniques to a wide range of biological and physical problems. The theoretical aspect involves translating algebraic information into dynamics and the numerical side involves computing these algebraic invariants by approximating the differential equations. The algebraic information works equally well for stable or unstable dynamics and, of course, being algebraic, does not need to be visualized in order to be understood.