Numerical Analysis of Qualitative Dynamics in Flows

研究者发展了确定常微分方程解的定性动力学行为的数值方法。对于这些方法，输入只需要一个微分方程系统，一个相空间区域，而不需要先验的动力学知识。输出是将不变集分解为包含非平凡动力学的区域的莫尔斯分解。这些区域的行为信息是根据康利指数给出的，并且可以检测到莫尔斯集之间的连接轨道。这种算法是研究微分方程组的有益的第一步，因为微分方程组的信息很少，特别是在高维中，数值模拟解的可视化是困难的。这里设想的计算类型提供了对动力系统的某种粗糙的定性描述。然而，康利指数，动力系统理论的一个工具，是一个代数拓扑指数，它也允许开发严格证明特定动力行为存在的算法。因此，一个可靠的算法既可以作为分析不熟悉系统的工具，也可以作为获得特定动力学的计算机辅助证明的工具；两者都有广泛的应用。这里开发的算法在计算拓扑中也有潜在的应用，特别是细胞复合体的同源性计算，以及与网格生成和Delaunay三角剖分的联系。微分方程被用来模拟物理科学中的许多现象：机械系统、化学反应和生物系统。理解这种固有非线性系统的动力学是全局分析中的一个问题。当底层空间的维数大于3时，对这种系统的研究往往超出了当前分析技术的范围。计算机能够对这类方程的解进行许多计算，但目前大多数数值方法都不太适合从高维系统中提取渐近动力学行为。一个问题是，计算机可以产生大量难以分析的数据；通常执行图形分析，这在高维中是不切实际的，并且可能错过不稳定的对象。本计画的目的是设计并实现研究高维微分方程渐近动力学的演算法。该算法的输出是一个代数索引（由C. Conley设计），可以将其视为从计算数据中提取基本动态信息（包括不稳定对象）的工具，并且不需要可视化就可以理解。

Kalies9973331The investigator develops numerical methods for determining the qualitative dynamical behavior of solutions to ordinary differential equations. For these methods, input requires only a system of differential equations, a region of the phase space, and no a priori knowledge of the dynamics. The output is a Morse decomposition of the invariant set into regions containing nontrivial dynamics. Information about the behavior in these regions is given in terms of the Conley index, and connecting orbits between Morse sets can be detected. Such an algorithm is beneficial as a first step in the study of systems of differential equations for which little information is known, particularly in higher dimensions where visualization of numerically simulated solutions is difficult. The type of computation envisioned here provides a somewhat coarse, qualitative description of a dynamical system. However, the Conley index, a tool from dynamical systems theory, is an algebraic topological index, which also allows for the development of algorithms for rigorously proving the existence of specific dynamical behavior. Therefore, a reliable algorithm can be used both as a tool for analyzing unfamiliar systems and as a tool for obtaining computer-assisted proofs of specific dynamics; both would have a wide variety of applications. The algorithms developed here also have potential applications in computational topology, in particular the computation of the homology of cellular complexes, and connections to mesh generation and Delaunay triangulations.Differential equations are used to model many phenomena in the physical sciences: mechanical systems, chemical reactions, and biological systems. Understanding the dynamics of such inherently nonlinear systems is a problem in global analysis. When the underlying space has dimension higher than three, the study of such systems is often beyond the scope of current analytic techniques. Computers are capable of performing many calculations of solutions to such equations, but most current numerical methods are not well suited for extracting asymptotic dynamical behavior from systems in higher dimensions. One problem is that computers can generate a large quantity of data that is difficult to analyze; often a graphical analysis is performed, which is impractical in higher dimensions and can miss unstable objects. The purpose of this project is to design and implement algorithms to study the asymptotic dynamics of differential equations in higher dimensions. The output of the algorithm is an algebraic index (designed by C. Conley) that could be thought of as a tool for extracting the essential dynamical information (including unstable objects) from computational data and that does not require visualization to be understood.